aa r X i v : . [ m a t h . G R ] N ov GROUPS ACTING ON TREESWITH PRESCRIBED LOCAL ACTION
STEPHAN TORNIER
Abstract.
We extend Burger–Mozes theory of closed, non-discrete, locallyquasiprimitive automorphism groups of locally finite, connected graphs to thesemiprimitive case, and develop a generalization of Burger–Mozes universalgroups acting on the regular tree T d of degree d . Three applications are given:First, we characterize the automorphism types which the quasi-center of a non-discrete subgroup of Aut( T d ) may feature in terms of the group’s local action.In doing so, we explicitly construct closed, non-discrete, compactly generatedsubgroups of Aut( T d ) with non-trivial quasi-center, and see that Burger–Mozestheory does not extend beyond the semiprimitive case to the transitive case.We then characterize the k -closures of locally transitive subgroups of Aut( T d ) containing an involutive inversion, and thereby partially answer two questionsby Banks–Elder–Willis. Finally, we offer a new view on the Weiss conjecture. Introduction
In the structure theory of locally compact (l.c.) groups, totally disconnected(t.d.) ones are in the focus because any locally compact group G is an extension ofits connected component G by the totally disconnected quotient G/G , / / G / / G / / G/G / / , and connected l.c. groups have been identified as inverse limits of Lie groups inseminal work by Gleason [Gle52], Montgomery-Zippin [MZ52] and Yamabe [Yam53].Every t.d.l.c. group can be viewed as a directed union of compactly generatedopen subgroups. Among the latter, groups acting on regular graphs and trees standout due to the Cayley-Abels graph construction: Every compactly generated t.d.l.c.group G acts vertex-transitively on a connected regular graph Γ of finite degree d with compact kernel K . In particular, the universal cover of Γ is the d -regular tree T d and we obtain a cocompact subgroup e G of its automorphism group Aut( T d ) , / / π (Γ) / / e G / / G/K / / , as an extension of π (Γ) by G/K , see [Mon01, Section 11.3] and [KM08] for details.In studying the automorphism group
Aut(Γ) of a locally finite, connected graph
Γ = (
V, E ) , we follow the notation of Serre [Ser03]. The group Aut(Γ) is t.d.l.c. whenequipped with the permutation topology for its action on V ∪ E , see Section 1.1.Given a subgroup H ≤ Aut(Γ) and a vertex x ∈ V , the stabilizer H x of x in H induces a permutation group on the set E ( x ) := { e ∈ E | o ( e ) = x } of edges issuingfrom x . We say that H is locally “P” if for every x ∈ V said permutation groupsatisfies property “P”, e.g. being transitive, semiprimitive or quasiprimitive.In [BM00], Burger–Mozes develop a remarkable structure theory of closed, non-discrete, locally quasiprimitive subgroups of Aut(Γ) , which resembles the theory ofsemisimple Lie groups, see Theorem 1.2. In Section 2, specifically Theorem 2.14,we show that this theory readily carries over to the semiprimitive case.
Date : November 19, 2020.
Let Ω be a set of cardinality d ∈ N ≥ and let T d = ( V, E ) be the d -regular tree.Burger–Mozes complement their structure theory with a particularly accessible classof subgroups of Aut( T d ) with prescribed local action: Given F ≤ Sym(Ω) theiruniversal group U( F ) is closed in Aut( T d ) , vertex-transitive, compactly generatedand locally permutation isomorphic to F . It is discrete if and only if F is semiregular.When F is transitive, U( F ) is maximal up to conjugation among vertex-transitivesubgroups of Aut( T d ) that are locally permutation isomorphic to F , hence universal .We generalize the universal groups by prescribing the local action on balls of agiven radius k ∈ N , the Burger–Mozes construction corresponding to the case k = 1 .Fix a tree B d,k which is isomorphic to a ball of radius k in the labelled tree T d andlet l kx : B ( x, k ) → B d,k ( x ∈ V ) be the unique label-respecting isomorphism. Then σ k : Aut( T d ) × V → Aut( B d,k ) , ( g, x ) → l kgx ◦ g ◦ ( l kx ) − captures the k -local action of g at the vertex x ∈ V . Definition
Let F ≤ Aut( B d,k ) . Define U k ( F ) := { g ∈ Aut( T d ) | ∀ x ∈ V : σ k ( g, x ) ∈ F } . While U k ( F ) is always closed, vertex-transitive and compactly generated, otherproperties of U( F ) require adjustments. Foremost, the group U k ( F ) need not belocally action isomorphic to F ; we say that F ≤ Aut( B d,k ) satisfies condition (C)if it is. This can be viewed as an interchangeability condition on neighbouring localactions, see Section 3.4. There also is a discreteness condition (D) on F ≤ Aut( B d,k ) in terms of certain stabilizers in F under which U k ( F ) is discrete, see Section 3.2.2.Finally, the groups U k ( F ) are universal in a sense akin to the above by Theorem 3.34.For e F ≤ Aut( B d,k ) , let F := π e F ≤ Sym(Ω) denote its projection to
Aut( B d, ) .The following rigidity theorem is inspired by [BM00, Proposition 3.3.1]. Theorem
Let F ≤ Sym(Ω) be -transitive and F ω ( ω ∈ Ω) simple non-abelian.Further, let e F ≤ Aut( B d,k ) with π e F = F satisfy (C). Then U k ( e F ) equals either U (Γ( F )) , U (∆( F )) or U ( F ) . The groups Γ( F ) , ∆( F ) ≤ Aut( B d, ) satisfy (C) and (D) and thus yield discreteuniversal groups. Illustrating the necessity of the assumptions, we construct otheruniversal groups when either point stabilizers in F are not simple, F is not primitive,or F is not perfect, see e.g. Φ( F, N ) , Φ( F, P ) , Π( F, ρ, X ) ≤ Aut( B d, ) in Section 3.4.In Section 4, we present three applications of the framework of universal groups.First, we study the quasi-center of subgroups of Aut( T d ) . The quasi-center QZ( G ) of a topological group G consists of those elements whose centralizer in G is open.It plays a major role in the Burger–Mozes Structure Theorem 1.2: A non-discrete,locally quasiprimitive subgroup of Aut( T d ) does not contain any non-trivial quasi-central elliptic elements. We complete this fact to the following local-to-global-typecharacterization of the automorphism types which the quasi-center of a non-discretesubgroup of Aut( T d ) may feature in terms of the group’s local action. Theorem
Let H ≤ Aut( T d ) be non-discrete. If H is locally(i) transitive then QZ( H ) contains no inversion.(ii) semiprimitive then QZ( H ) contains no non-trivial edge-fixating element.(iii) quasiprimitive then QZ( H ) contains no non-trivial elliptic element.(iv) k -transitive ( k ∈ N ) then QZ( H ) contains no hyperbolic element of length k .More importantly, the proof of the above theorem suggests to use groups ofthe form T k ∈ N U k ( F ( k ) ) for appropriate local actions F ( k ) in order to explicitly construct non-discrete subgroups of Aut( T d ) whose quasi-centers contain certaintypes of elements. This leads to the following sharpness result. ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 3
Theorem
There is d ∈ N ≥ and a closed, non-discrete, compactly generatedsubgroup of Aut( T d ) which is locally(i) intransitive and contains a quasi-central inversion.(ii) transitive and contains a non-trivial quasi-central edge-fixating element.(iii) semiprimitive and contains a non-trivial quasi-central elliptic element.(iv) (a) intransitive and contains a quasi-central hyperbolic element of length .(b) quasiprimitive and contains a quasi-central hyperbolic element of length .Part (ii) of this theorem can be strengthened to the following result which showsthat Burger–Mozes theory does not extend further to locally transitive groups. Theorem
There is d ∈ N ≥ and a closed non-discrete, compactly generated,locally transitive subgroup of Aut( T d ) with open, hence non-discrete, quasi-center.Second, we give an algebraic characterization of the k -closures of locally transitivesubgroups of Aut( T d ) which contain an involutive inversion, and thereby partiallyanswer two question by Banks–Elder–Willis [BEW15, p. 259]. Recall (Section 1.2)that the k -closure ( k ∈ N ) of a subgroup H ≤ Aut( T d ) is given by H ( k ) = { g ∈ Aut( T d ) | ∀ x ∈ V ( T d ) ∃ h ∈ H : g | B ( x,k ) = h | B ( x,k ) } . Theorem
Let H ≤ Aut( T d ) be locally transitive and contain an involutiveinversion. Then H ( k ) = U k ( F ( k ) ) for some labelling l of T d and F ( k ) ≤ Aut( B d,k ) .Combined with the independence properties P k ( k ∈ N ) (Section 1.2), introducedby Banks–Elder–Willis [BEW15] as generalizations of Tits’ Independence Property,Theorem 4.31 entails the following characterization of universal groups. Corollary
Let H ≤ Aut( T d ) be closed, locally transitive and contain aninvolutive inversion. Then H = U k ( F ( k ) ) if and only if H satisfies Property P k .Banks–Elder–Willis use subgroups of Aut( T d ) with pairwise distinct k -closuresto construct infinitely many, pairwise non-conjugate, non-discrete simple subgroupsof Aut( T d ) via Theorem 1.1 and ask whether they are also pairwise non-isomorphicas topological groups. We partially answer this question in the following theorem. Theorem
Let H ≤ Aut( T d ) be non-discrete, locally permutation isomorphicto F ≤ Sym(Ω) and contain an involutive inversion. Suppose that F is transitiveand that every non-trivial subnormal subgroup of F ω ( ω ∈ Ω) is transitive on Ω \{ ω } .If H ( k ) = H ( l ) for some k, l ∈ N then ( H ( k ) ) + k and ( H ( l ) ) + l are non-isomorphic.Infinitely many families of pairwise non-isomorphic simple groups of this type,each sharing a certain transitive local action, are constructed in Example 4.37.Finally, Section 4.3 offers a new view on the Weiss conjecture [Wei78] which statesthat there are only finitely many conjugacy classes of discrete, locally primitive andvertex-transitive subgroups of Aut( T d ) for a given d ∈ N ≥ : Under the additionalassumption that each group contains an involutive inversion, it suffices to showthat for every primitive F ≤ Sym(Ω) there are only finitely many e F ≤ Aut( B d,k )( k ∈ N ) with π e F = F that satisfy conditions (C) and (D) in a minimal fashion. Acknowledgements.
The author is indebted to Marc Burger and George Willisfor their support and the suggestion to define generalized universal groups. Thanksare also due to Luke Morgan and Michael Giudici for sharing their insight onpermutation groups, and Michael Giudici, for providing a proof of Lemma 3.29. Ananonymous referee’s comments were also much appreciated.A good part of this research was carried out during visits to The University ofNewcastle, Australia, for the hospitality of which the author is thankful. Finally,part of this research was supported by the SNSF Doc.Mobility fellowship 172120and the ARC Discovery Project 120100996 which are gratefully acknowledged.
STEPHAN TORNIER Preliminaries
This section collects preliminaries on permutation groups, graph theory andBurger–Mozes theory. References are given in the respective subsection.1.1.
Permutation Groups.
Let Ω be a set. In this section, we collect definitionsand results concerning Sym(Ω) , the group of bijections of Ω . Refer to [DM96],[Pra97] and [GM18] for details beyond the following.Let F ≤ Sym(Ω) . The degree of F is | Ω | . For ω ∈ Ω , the stabilizer of ω in F is F ω := { σ ∈ F | σω = ω } . The subgroup of F generated by its point stabilizersis denoted by F + := h{ F ω | ω ∈ Ω }i . The permutation group F is semiregular , or free , if F ω = { id } for all ω ∈ Ω ; equivalently, if F + is trivial. It is transitive if itsaction on Ω is transitive, and regular if it is both semiregular and transitive.Let F ≤ Sym(Ω) be transitive. The rank of F is the number rank( F ) := | F \ Ω | of orbits of the diagonal action σ · ( ω, ω ′ ) := ( σω, σω ′ ) of F on Ω . Equivalently, rank( F ) = | F ω \ Ω | for all ω ∈ Ω . Note that the diagonal ∆(Ω) := { ( ω, ω ) | ω ∈ Ω } is always an orbit of the diagonal action F y Ω . The permutation group F is -transitive if it acts transitively on Ω \ ∆(Ω) . In other words, rank( F ) = 2 .We now define several classes of permutation groups lying in between the classesof transitive and -transitive permutation groups. Let F ≤ Sym(Ω) . A partition P : Ω = F i ∈ I Ω i of Ω is preserved by F , or F -invariant , if for all σ ∈ F we have { σ Ω i | i ∈ I } = { Ω i | i ∈ I } . The partition of Ω as Ω itself, as well as the partitionof Ω into singletons, is trivial . A map a : Ω → F is constant with respect to P if a ( ω ) = a ( ω ′ ) whenever ω, ω ′ ∈ Ω i for some i ∈ I .Let F ≤ Sym(Ω) . The permutation group F is primitive if it is transitive andpreserves no non-trivial partition of Ω . Equivalently, F is transitive and its pointstabilizers are maximal subgroups. It is imprimitive otherwise. Given a normalsubgroup N of F , the partition of Ω into N -orbits is F -invariant. Consequently,every non-trivial normal subgroup of a primitive group is transitive. A permutationgroup is quasiprimitive if it is transitive and all its non-trivial normal subgroupsare transitive. Finally, a permutation group is semiprimitive if it is transitive andall its normal subgroups are either transitive or semiregular. The following chain ofimplications among properties of permutation groups follows from the definitions.We list examples illustrating that each implication is strict. -transitive ⇒ primitive A y A /D ⇒ quasiprimitive A y A /C ⇒ semiprimitive C D C ⇒ transitive D D C × C Note that every simple transitive group is quasiprimitive, and that C (cid:12) D (cid:12) A is a non-maximal subgroup. Permutation Topology.
Let X be a set and H ≤ Sym( X ) . The basic open sets ofthe permutation topology on H are U x,y := { h ∈ H | ∀ i ∈ { , . . . , n } : h ( x i ) = y i } ,where n ∈ N and x = ( x , . . . , x n ) , y = ( y , . . . , y n ) ∈ X n . This topology turns H into a Hausdorff, totally disconnected topological group and makes the action map H × X → X continuous, where X is equipped with the discrete topology. Note that Sym( X ) is second-countable if and only if X is countable. See [KM08, Section 1.2].1.2. Graph Theory.
We first recall Serre’s [Ser03] notation and definitions in thecontext of graphs and trees, and then collect generalities about automorphisms oftrees. We conclude with an important simplicity criterion.
Definitions and Notation. A graph Γ is a tuple ( V, E ) consisting of a vertex set V and an edge set E , together with a fixed-point-free involution of E , denoted by e e , and maps o, t : E → V , providing the origin and terminus of an edge, suchthat o ( e ) = t ( e ) and t ( e ) = o ( e ) for all e ∈ E . Given e ∈ E , the pair { e, e } is a ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 5 geometric edge . For x ∈ V , we let E ( x ) := o − ( x ) = { e ∈ E | o ( e ) = x } be theset of edges issuing from x . The valency of x ∈ V is | E ( x ) | . A vertex of valency is a leaf . A morphism between graphs Γ = ( V , E ) and Γ = ( V , E ) is a pair ( α V , α E ) of maps α V : V → V and α E : E → E preserving the graph structure,i.e. α V ( o ( e )) = o ( α E ( e )) and α V ( t ( e )) = t ( α E ( e )) for all e ∈ E .For n ∈ N , let Path n denote the graph with vertex set { , . . . , n } and edge set { ( k, k + 1) , ( k, k + 1) | k ∈ { , . . . , n − }} . A path of length n in a graph Γ is amorphism γ from Path n to Γ . It can be identified with ( e , . . . , e n ) ∈ E (Γ) n , where e k is the image of ( k − , k ) ∈ E (Path n ) for all k ∈ { , . . . , n } . In this case, γ is apath from o ( e ) to t ( e n ) .Similarly, let Path N and Path Z denote the graphs with vertex sets N and Z ,and edge sets { ( k, k + 1) , ( k, k + 1) | k ∈ N } and { ( k, k + 1) , ( k, k + 1) | k ∈ Z } re-spectively. A half-infinite path, or ray , in a graph Γ is a morphism γ from Path N to Γ . It can be identified with ( e k ) k ∈ N ∈ E (Γ) N where e k = γ ( k − , k ) for all k ∈ N .In this case, γ originates at , or issues from , o ( e ) . An infinite path , or line , in agraph Γ is a morphism γ from Path Z to Γ .A pair ( e k , e k +1 ) = ( e k , e k ) in a path is a backtracking . A graph is connected ifany two of its vertices can be joined by a path. The maximal connected subgraphsof a graph are its connected components .A forest is a graph in which there are no non-backtracking paths ( e , . . . , e n ) with o ( e ) = t ( e n ) ( n ∈ N ) . Consequently, a morphism of forests is determinedby the underlying vertex map. In particular, a path of length n ∈ N in a forest isdetermined by the images of the vertices of Path n .A tree is a connected forest. As a consequence of the above, the vertex set V of a tree T admits a natural metric: Given x, y ∈ V , define d ( x, y ) as the minimallength of a path from x to y . A tree in which every vertex has valency d ∈ N is d -regular . It is unique up to isomorphism and denoted by T d .Let T = ( V, E ) be a tree. For S ⊆ V ∪ E , the subtree spanned by S is theunique minimal subtree of T containing S . For x ∈ V and n ∈ N , the subtreespanned by { y ∈ V | d ( y, x ) ≤ n } is the ball of radius n around x , denoted by B ( x, n ) . Similarly, S ( x, n ) = { y ∈ V | d ( y, x ) = n } is the sphere of radius n around x , and E ( x, n ) := { e ∈ E | d ( o ( e ) , x ) , d ( t ( e ) , x ) ≤ n } . For a subtree T ′ ⊆ T ,let π : V → V ( T ′ ) denote the closest point projection, i.e. π ( x ) = y whenever d ( x, y ) = min z ∈ V ( T ′ ) { d ( x, z ) } . In the case of an edge e = ( v, w ) ∈ E , the half-trees T v and T w are the subtrees spanned by π − ( v ) and π − ( w ) respectively.Two rays γ , γ : Path N → T in T are equivalent , γ ∼ γ , if there exist N, d ∈ N such that γ ( n ) = γ ( n + d ) for all n ≥ N . The boundary , or set of ends , of T is theset ∂T of equivalence classes of rays in T . Automorphism Groups of Graphs.
Let
Γ = (
V, E ) be a graph. The group Aut(Γ) of automorphims of Γ is our foremost concern. Throughout, we equip Aut(Γ) withthe permutation topology for its action on V ∪ E .Notation. Let H ≤ Aut(Γ) . Given a subgraph Γ ′ ⊆ Γ , the pointwise stabilizer of Γ ′ in H is denoted by H Γ ′ . Similary, the setwise stabilizer of Γ ′ in H is denotedby H { Γ ′ } . In the case where Γ ′ is a single vertex x , the permutation group that H x induces on E ( x ) is denoted by H (1) x ≤ Sym( E ( x )) . Given a property “P” ofpermutation groups, the group H is locally “P” if for every x ∈ V the permutationgroup H (1) x has “P”; with the exception that H is locally k -transitive ( k ∈ N ≥ ) if H x acts transitively on the set of non-backtracking paths of length k issuing from x .It is locally ∞ -transitive if it is locally k -transitive for all k ∈ N .Now, let d ∈ N ≥ and T d = ( V, E ) the d -regular tree. The group Aut( T d ) actson ∂T d by g · [ γ ] := [ g ◦ γ ] . Given an end [ γ ] ∈ ∂T d , the stabilizer of [ γ ] in H is H [ γ ] = { h ∈ H | h ◦ γ ∼ γ } . STEPHAN TORNIER
We let H + = h{ H x | x ∈ V }i denote the subgroup of H generated by vertex-stabilizers and H + = h{ H e | e ∈ E }i the subgroup generated by edge-stabilizers.For a subtree T ⊆ T d and k ∈ N , let T k denote the subtree of T d spanned by { x ∈ V | d ( x, T ) ≤ k } . We set H + k = h{ H e k − | e ∈ E }i . Then H + = H + and H + k E H + E H + E H. Classification of Automorphisms. Automorphisms of T d can be distinguished intothree distinct types. Refer to [GGT18, Section 6.2.2] for details.For g ∈ Aut( T d ) , set l ( g ) := min x ∈ V d ( x, gx ) and V ( g ) := { x ∈ V | d ( x, gx ) = l ( g ) } .If l ( g ) = 0 then g fixes a vertex. An automorphism of this kind is elliptic . Supposenow that l ( g ) > . If V ( g ) is infinite then g is hyperbolic . Geometrically, it is atranslation of length l ( g ) along the line in T d defined by V ( g ) . If V ( g ) is finite then l ( g ) = 1 and g maps some edge e ∈ E to e , and is termed an inversion .Independence and Simplicity. The base case of the simplicity criterion presented inthis paragraph is due to Tits [Tit70] and applies to sufficiently large subgroups of Aut( T d ) with a certain independence property. The generalized version is due toBanks–Elder–Willis [BEW15], see also [GGT18].Let C denote a path in T d (finite, half-infinite or infinite). For every x ∈ V ( C ) and k ∈ N , the pointwise stabilizer H C k of C k induces an action H ( x ) C k ≤ Aut( π − ( x )) on π − ( x ) , the subtree spanned by those vertices of T whose closest vertex in C is x . We therefore obtain an injective homomorphism ϕ ( k ) C : H C k → Y x ∈ V ( C ) H ( x ) C k . A subgroup H ≤ Aut( T d ) satisfies Property P k ( k ∈ N ) if ϕ ( k − C is an isomorphismfor every path C in T d . If H ≤ Aut( T d ) is closed, it suffices to check the aboveproperties in the case where C is a single edge. For example, given a closed subgroup H ≤ Aut( T d ) , Property P ( k ) is satisfied by its k -closure H ( k ) = { g ∈ Aut( T d ) | ∀ x ∈ V ( T d ) ∃ h ∈ H : g | B ( x,k ) = h | B ( x,k ) } . Theorem 1.1 ([BEW15, Theorem 7.3]) . Let H ≤ Aut( T d ) . Suppose H neitherfixes an end nor stabilizes a proper subtree of T d setwise, and that H satisfiesProperty P k . Then the group H + k is either trivial or simple. Burger–Mozes Theory.
In [BM00], Burger–Mozes develop a structure theory ofcertain locally quasiprimitive automorphism groups of graphs which resembles thetheory of semisimple Lie groups. Their fundamental definitions are meaningful inthe setting of t.d.l.c. groups. Let H be a t.d.l.c. group. Define H ( ∞ ) := \ { N E H | N is closed and cocompact in H } , alternatively the intersection of all open finite-index subgroups of H , and QZ( H ) := { h ∈ H | Z H ( h ) ≤ H is open } , the quasi-center of H . Both H ( ∞ ) and QZ( H ) are topologically characteristic sub-groups of H , i.e. they are preserved by continuous automorphisms of H . Whereas H ( ∞ ) ≤ H is closed, the quasi-center need not be so.In p -adic semisimple algebraic groups, H ( ∞ ) and QZ( H ) play roles analogous tothe connected component of the identity and the kernel of the adjoint representationas [BM00, Example 1.1.1.] shows.Whereas for a general t.d.l.c. group H nothing much can be said about the sizeof H ( ∞ ) and QZ( H ) , Burger–Mozes show that good control can be obtained in thecase of certain locally quasiprimitive automorphism groups of graphs. The followingresult summarizes their structure theory. It is a combination of Proposition 1.2.1,Corollary 1.5.1, Theorem 1.7.1 and Corollary 1.7.2 in [BM00]. ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 7
Theorem 1.2.
Let Γ be a locally finite, connected graph. Further, let H ≤ Aut(Γ) be closed, non-discrete and locally quasiprimitive. Then(i) H ( ∞ ) is minimal closed normal cocompact in H ,(ii) QZ( H ) is maximal discrete normal, and non-cocompact in H , and(iii) H ( ∞ ) / QZ( H ( ∞ ) ) = H ( ∞ ) / (QZ( H ) ∩ H ( ∞ ) ) admits minimal, non-trivial closednormal subgroups; finite in number, H -conjugate and topologically simple.If Γ is a tree, and, in addition, H is locally primitive then(iv) H ( ∞ ) / QZ( H ( ∞ ) ) is a direct product of topologically simple groups. Burger–Mozes Universal Groups.
The first introduction of Burger–Mozes universalgroups in [BM00, Section 3.2] was expanded in the introductory article [GGT18],which we follow closely. Most results are generalized in Section 3.Let Ω be a set of cardinality d ∈ N ≥ and let T d = ( V, E ) denote the d -regulartree. A labelling l of T d is a map l : E → Ω such that for every x ∈ V the map l x : E ( x ) → Ω , y l ( y ) is a bijection, and l ( e ) = l ( e ) for all e ∈ E . The local action σ ( g, x ) ∈ Sym(Ω) of an automorphism g ∈ Aut( T d ) at a vertex x ∈ V is defined via σ : Aut( T d ) × X → Sym(Ω) , ( g, x ) σ ( g, x ) := l gx ◦ g ◦ l − x . Definition 1.3.
Let F ≤ Sym(Ω) and l a labelling of T d . Define U ( l ) ( F ) := { g ∈ Aut( T d ) | ∀ x ∈ V : σ ( g, x ) ∈ F } . The map σ satisfies a cocycle identity : For all g, h ∈ Aut( T d ) and x ∈ V we have σ ( gh, x ) = σ ( g, hx ) σ ( h, x ) . As a consequence, U ( l ) ( F ) is a subgroup of Aut( T d ) .Passing to a different labelling amounts to passing to a conjugate of U ( l ) ( F ) inside Aut( T d ) . We therefore omit the reference to an explicit labelling from here on.The following proposition collects several basic properties of Burger–Mozes groups.We refer the reader to [GGT18, Section 6.4] for proofs. Proposition 1.4.
Let F ≤ Sym(Ω) . The group U( F ) is(i) closed in Aut( T d ) ,(ii) vertex-transitive,(iii) compactly generated,(iv) locally permutation isomorphic to F ,(v) edge-transitive if and only if F is transitive, and(vi) discrete if and only if F is semiregular.Part (iii) of Proposition 1.4 relies on the following result which we include forfuture reference. Given x ∈ V and ω ∈ Ω , let ι ( x ) ω ∈ U( { id } ) denote the uniquelabel-respecting inversion of the edge e ω ∈ E with o ( e ω ) = x and l ( e ω ) = ω . Lemma 1.5.
Let x ∈ V . Then U( { id } ) = h{ ι ( x ) ω | ω ∈ Ω }i ∼ = ∗ ω ∈ Ω h ι ( x ) ω i ∼ = ∗ ω ∈ Ω Z / Z . Proof.
Every element of U( { id } ) is determined by its image on x . Hence it sufficesto show that h{ ι ( x ) ω | ω ∈ Ω }i is vertex-transitive and has the asserted structure.Indeed, let y ∈ V \{ x } , and let ω , . . . , ω n ∈ Ω be the labels of the geodesic from x to y . Then ι ( x ) ω ◦ · · · ◦ ι ( x ) ω n maps x to y as every ι ( x ) ω ( ω ∈ Ω) is label-respecting.Setting X ω := T t ( e ω ) we have ι ω ( X ω ′ ) ⊆ X ω for all distinct ω, ω ′ ∈ Ω . Hence theassertion follows from the ping-pong lemma. (cid:3) The name universal group is due to the following maximality statement. Its proof,see [GGT18, Proposition 6.23], should be compared with the proof of Theorem 3.34.
Proposition 1.6.
Let H ≤ Aut( T d ) be locally transitive and vertex-transitive.Then there is a labelling l of T d such that H ≤ U ( l ) ( F ) where F ≤ Sym(Ω) isaction isomorphic to the local action of H . STEPHAN TORNIER Structure Theory of locally semiprimitive groups
We generalize the Burger–Mozes theory of locally quasiprimitive automorphismgroups of graphs to the semiprimitive case. While this adjustment of Sections 1.1to 1.5 in [BM00] is straightforward and has been initiated in [Tor18, Section II.7]and [CB19, Section 6.2] we provide a full account for the reader’s convenience.2.1.
General Facts.
Let
Γ = (
V, E ) be a connected graph. We first collect a fewgeneral facts about several classes of subgroups of Aut(Γ) for future reference.
Lemma 2.1.
Let H ≤ Aut(Γ) be locally transitive. Then H + is geometric edgetransitive and of index at most in H . Proof.
Since H is locally transitive, so is H + given that H + x = H x for all x ∈ V .Hence it is geometric edge transitive. In particular it has at most two vertex orbitswhich implies the second assertion. (cid:3) Lemma 2.2.
Let H ≤ Aut(Γ) and let Γ ′ = ( V ′ , E ′ ) be a connected subgraph of Γ .Suppose R ⊆ H is such that for every x ′ ∈ V ′ and e ∈ E ( x ′ ) there is r ∈ R suchthat re ∈ E ′ . Then Λ := h R i satisfies S λ ∈ Λ λ Γ ′ = Γ . Proof.
By assumption, B (Γ ′ , ⊆ S λ ∈ Λ λ Γ ′ . Now suppose B (Γ ′ , n ) ⊆ S λ ∈ Λ λ Γ ′ forsome n ∈ N . Let x ′ ∈ V ( B (Γ ′ , n )) . Pick λ ∈ Λ such that λ ( x ′ ) ∈ V ′ . Since λ induces abijection between E ( x ′ ) and E ( λ ( x ′ )) we conclude that B (Γ ′ , n +1) ⊆ S λ ∈ Λ λ Γ ′ . (cid:3) Assume from now on that Γ is a locally finite, connected graph. Lemma 2.3.
Let H ≤ Aut(Γ) . If H \ Γ is finite then there is a finitely generatedsubgroup Λ ≤ H such that Λ \ Γ is finite. Proof.
Let Γ ′ = ( V ′ , E ′ ) ⊆ Γ be a connected subgraph which projects onto H \ Γ .For every x ′ ∈ V ′ and e ∈ E ( x ′ ) , pick λ x ′ ,e ∈ H such that λ x ′ ,e ( e ) ∈ E ′ . Then Λ := h{ λ x ′ ,e | x ′ ∈ X, e ∈ E ( x ′ ) }i satisfies the conclusion by Lemma 2.2. (cid:3) Lemma 2.4.
Let Λ ≤ Aut(Γ) . If Λ \ Γ is finite then Z Aut(Γ) (Λ) is discrete.
Proof.
Let F ⊆ E be finite such that S λ ∈ Λ λF = E and U := Λ F ∩ Z Aut(Γ) (Λ) ,which is open in Z Aut(Γ) (Λ) . Given that U and Λ commute, U acts trivially on E = S λ ∈ Λ λF . Hence U = { id } and Z Aut(Γ) (Λ) is discrete. (cid:3)
Lemma 2.5.
Let Λ , Λ ≤ Aut(Γ) . If Λ \ Γ is finite and [Λ , Λ ] ≤ Aut(Γ) isdiscrete then Λ ≤ Aut(Γ) is discrete.
Proof.
Using Lemma 2.3 pick R ⊆ Λ such that h R i\ Γ is finite. As [Λ , Λ ] ≤ Aut(Γ) is discrete, there is an open subgroup U ≤ Λ such that [ r, U ] = { e } for all r ∈ R .That is U ≤ Z Aut(Γ) ( h R i ) . Hence U is discrete by Lemma 2.4, and so is Λ . (cid:3) Lemma 2.6.
Let H ≤ Aut(Γ) be non-discrete. Then
QZ( H ) \ Γ is infinite. Proof. If QZ( H ) \ Γ is finite, there is a finitely generated subgroup Λ \ QZ( H ) suchthat Λ \ Γ is finite as well by Lemma 2.3. Hence there is an open subgroup U ≤ H with U ≤ Z Aut(Γ) (Λ) . Hence U and thereby H is discrete. (cid:3) Lemma 2.7.
Let Λ ≤ Aut(Γ) be discrete. If Λ \ Γ is finite then N Aut(Γ) (Λ) is discrete.
Proof.
Apply Lemma 2.5 to Λ := Λ and Λ := N Aut(Γ) (Λ) . (cid:3) ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 9
Normal Subgroups.
Let
Γ = (
V, E ) denote a locally finite, connected graph.For closed subgroups Λ E H of Aut(Γ) we define N nf ( H, Λ) = { N E H | Λ ≤ N E H, N is closed and does not act freely on E } , the set of closed normal subgroups of H which contain Λ and do not act freely on E .The set N nf ( H, Λ) is partially ordered by inclusion. We let M nf ( H, Λ) ⊆ N nf ( H, Λ) denote the set of minimal elements in N nf ( H, Λ) . Lemma 2.8.
Let
Γ = (
V, E ) be a locally finite, connected graph and Λ E H ≤ Aut(Γ) .If H \ Γ is finite and H does not act freely on E then M nf ( H, Λ) = ∅ . Proof.
We argue using Zorn’s Lemma. First note that N nf ( H, Λ) is non-empty as itcontains H . Let C ⊆ N nf ( H, Λ) be a chain. Pick a finite set F ⊆ E of representativesof H \ E . For every N ∈ C , the set F N := { e ∈ F | N | e ≤ Aut( e ) is non-trivial } isnon-empty. Since F is finite and C is a chain it follows that T N ∈ C F N is non-empty,i.e. there exists e ∈ F such that N | e is non-trivial for every N ∈ C . As before, weconclude that M := T N ∈ C N | e is non-trivial. Now, for α ∈ M \{ id } and N ∈ C ,the set N α := { g ∈ N e | g | e = α } is a non-empty compact subset of H e , and since C is a chain every finite subset of { N α | N ∈ C } has non-empty intersection. Hence T N ∈ C N α is non-empty and therefore N C := T N ∈ C N is a closed normal subgroupof H containing Λ that does not act freely on E . Overall, N C ∈ M nf ( H, Λ) . (cid:3) The following lemma is contained in the author’s PhD thesis [Tor18, Section II.7]and, independently, in Caprace-Le Boudec [CB19, Section 6.2].
Lemma 2.9.
Let
Γ = (
V, E ) be a locally finite, connected graph. Further, let H ≤ Aut(Γ) be locally semiprimitive and N E H . Define V := { x ∈ V | N x y S ( x, is transitive and not semiregular } , V := { x ∈ V | N x y S ( x, is semiregular } .Then one of the following holds.(i) V = V and N acts freely on E .(ii) V = V and N is geometric edge transitive.(iii) V = V ⊔ V is an H -invariant partition of V and B ( x, is a fundamentaldomain for the action of N on Γ for any x ∈ V . Proof.
Since H is locally semiprimitive and N is normal in H , we have V = V ⊔ V .If N does not act freely on E then there is an edge e ∈ E with N e = { id } and an N e -fixed vertex x ∈ V for which N x y S ( x, is not semiregular, hence transitive.That is, V = ∅ . Now, either V ( N ) = ∅ in which case N is locally transitive andwe are in case (ii), or V ( N ) = ∅ . Being locally transitive, H acts transitively onthe set of geometric edges and therefore has at most two vertex orbits. Given thatboth V and V are non-empty and H -invariant, they constitute exactly said orbits.Since any pair of adjacent vertices ( x, y ) is a fundamental domain for the H -actionon V , we conclude that if y ∈ V then x ∈ V . Thus every leaf of B ( y, is in V and we are in case (iii) by Lemma 2.2. (cid:3) The Subquotient H ( ∞ ) / QZ( H ( ∞ ) . In this section, we achieve control over H ( ∞ ) and QZ( H ) as well as the normal subgroups of H in the semiprimitive case.We then describe the structure of the subquotient H ( ∞ ) / QZ( H ( ∞ ) ) . First, recallthe following lemma from topological group theory. Lemma 2.10.
Let G be a topological group. If H E G is discrete then H ⊆ QZ( G ) . Proof.
For h ∈ H , the map ϕ h : G → H , g ghg − is well-defined because H E G ,and continuous. Hence there is an open set U ⊆ G containing ∈ G and such that ϕ h ( U ) ⊆ { h } , i.e. U ⊆ Z G ( h ) . (cid:3) Proposition 2.11.
Let
Γ = (
V, E ) be a locally finite, connected graph. Further,let H ≤ Aut(Γ) be closed, non-discrete and locally semiprimitive. Then(i)
H/H ( ∞ ) is compact,(ii) QZ( H ) acts freely on E , and is discrete non-cocompact in H ,(iii) for any closed normal subgroup N E H , either N is non-discrete cocompactand N D H ( ∞ ) , or N is discrete and N E QZ( H ) ,(iv) QZ( H ( ∞ ) ) = QZ( H ) ∩ H ( ∞ ) acts freely on E without inversions,(v) for any open normal subgroup N E H ( ∞ ) we have N = H ( ∞ ) , and(vi) H ( ∞ ) is topologically perfect, i.e. H ( ∞ ) = [ H ( ∞ ) , H ( ∞ ) ] . Proof.
For (i), let N E H be closed and cocompact. Since H is non-discrete, so is N inview of Lemma 2.7. Hence N ∈ N nf ( H, { id } ) . Conversely, if N ∈ N nf ( H, { id } ) then N is cocompact in H by Lemma 2.9. We conclude that H ( ∞ ) = T N nf ( H, { id } ) .This intersection is in fact given by a single minimal element of N nf ( H, { id } ) : UsingLemma 2.8, pick M ∈ M nf ( H, { id } ) , and let N ∈ N nf ( H, { id } ) . Suppose N M .Because M is minimal, N ∩ M acts freely on E . In particular, N ∩ M is discrete.Since both N and M are normal in H , we also have N ∩ M ⊇ [ N, M ] and hence N and M are discrete by Lemma 2.5. Then so is H ⊆ N Aut( g ) ( H ) by Lemma 2.7.Overall, H ( ∞ ) = M ∈ M nf ( H, { id } ) and assertion now follows from Lemma 2.9.As to (ii), the group QZ( H ) is non-cocompact by Lemma 2.6 and therefore actsfreely on E by Lemma 2.9. In particular, it is discrete.For (iii), let N E H be a closed normal subgroup. If N acts freely on E , then N is discrete and hence contained in QZ( H ) by Lemma 2.10. If N does not act freelyon E then N is cocompact in H by Lemma 2.9 and therefore contains H ( ∞ ) .Concerning (iv) the inclusion QZ( H ) ∩ H ( ∞ ) ⊆ QZ( H ( ∞ ) ) is automatic. Further, QZ( H ( ∞ ) ) is normal in H because it is topologically characteristic in H ( ∞ ) E H .Therefore, if QZ( H ( ∞ ) ) QZ( H ) , then QZ( H ( ∞ ) ) is non-discrete by part (iii) anddoes not act freely on E . Then QZ( H ( ∞ ) ) \ Γ is finite by Lemma 2.9, contradictingLemma 2.6 applied to H ( ∞ ) which is non-discrete because QZ( H ( ∞ ) ) ≤ H ( ∞ ) is.Consequently, QZ( H ( ∞ ) ) ≤ QZ( H ) which proves the assertion.For part (v), note that M nf ( H ( ∞ ) , { id } ) is non-empty by Lemma 2.8 as H ( ∞ ) is cocompact in Aut(Γ) by part (i) and non-discrete by part (iii). Further, since
QZ( H ( ∞ ) ) acts freely on E , every N ∈ N nf ( H ( ∞ ) , { id } ) is non-discrete by part (iii)as well. Given an open subgroup U E H ( ∞ ) and N ∈ M nf ( H ∞ , { id } ) , the group U ∩ N is normal in H ( ∞ ) and non-discrete. In particular, U ∩ N does not act freelyon E and hence U ∩ N = N . Thus U contains the subgroup of H ( ∞ ) generated bythe elements of M nf ( H ( ∞ ) , { id } ) , which is closed, normal and non-discrete. Hence U = H ( ∞ ) .As to (vi), the group [ H ( ∞ ) , H ( ∞ ) ] is non-discrete by part (i) and Lemma 2.5.Hence so is [ H ( ∞ ) , H ( ∞ ) ] E H ( ∞ ) . Now apply part (iii). (cid:3) Proposition 2.12.
Let
Γ = (
V, E ) be a locally finite, connected graph. Further, let H ≤ Aut(Γ) be a closed, non-discrete and locally semiprimitive. Finally, let Λ E H such that Λ ≤ QZ( H ( ∞ ) ) . Then the following hold.(i) (a) The group H acts transitively on M nf ( H ( ∞ ) , Λ) .(b) The set M nf ( H ( ∞ ) , Λ) is finite.(ii) Let M ∈ M nf ( H ( ∞ ) , Λ) (a) The group M/ Λ is topologically perfect.(b) The group QZ( M ) acts freely on E and QZ( M ) = QZ( H ( ∞ ) ) ∩ M .(c) The group M/ QZ( M ) is topologically simple.(iii) For every N ∈ N nf ( H ( ∞ ) , Λ) there is M ∈ M nf ( H ( ∞ ) , Λ) with N ⊇ M . ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 11
Proof.
Since every discrete normal subgroup of H ( ∞ ) is contained in QZ( H ( ∞ ) ) byLemma 2.10 (iii) and the latter acts freely on E by Proposition 2.11 (iii), everyelement of N nf ( H ( ∞ ) , Λ) is non-discrete. We proceed with a number of claims.(1) For every N ∈ N nf ( H ( ∞ ) , Λ) we have [ H ( ∞ ) , N ] QZ( H ( ∞ ) ) .This follows from the above combined with 2.11 (i) and Lemma 2.5.In the following, given S ⊆ M nf ( H ( ∞ ) , Λ) , we let M S := h M | M ∈ S i ≤ H ( ∞ ) denote the subgroup of H ( ∞ ) generated by S M ∈ S M .(2) The group H acts transitively on M nf ( H ( ∞ ) , Λ) .Let S be an orbit for the action of H on M nf ( H ( ∞ ) , Λ) , and suppose there isan element M ∈ M nf ( H ( ∞ ) , Λ) \ S . For every N ∈ S , the subgroup N ∩ M isnormal in H ( ∞ ) and acts freely on E by minimality of M , hence is discrete.The same therefore holds for [ N, M ] ⊆ N ∩ M . Thus [ N, M ] ⊆ QZ( H ( ∞ ) ) .As QZ( H ( ∞ ) ) is discrete by Proposition 2.11 and therefore closed in H ( ∞ ) we conclude [ M S , M ] ⊆ QZ( H ( ∞ ) ) . On the other hand, M S is normal in H since S is an H -orbit. It is also closed in H , and non-discrete by the above.Thus M S = H ( ∞ ) by Proposition 2.11 (iii), and [ H ( ∞ ) , M ] ⊆ QZ( H ( ∞ ) ) which contradicts part (1).(3) For every M ∈ M nf ( H ( ∞ ) , Λ) we have [ M, M ] · Λ = M Note that [ M, M ] · Λ is a group because Λ is normal in M . Suppose there isan element M ∈ M nf ( H ( ∞ ) , Λ) with [ M , M ] · Λ (cid:12) M . Then [ M , M ] · Λ acts freely on E by minimality of M and is discrete. Being normal in H , weobtain [ M , M ] ⊆ QZ( H ( ∞ ) . Part (2) now implies that [ M, M ] ⊆ QZ( H ( ∞ ) ) for all M ∈ M nf ( H ( ∞ ) , Λ) . Given that [ M, M ′ ] ⊆ QZ( H ( ∞ ) ) for all distinct M, M ′ in M nf ( H ( ∞ ) , Λ) as well, we conclude that [ H ( ∞ ) , H ( ∞ ) ] ⊆ QZ( H ( ∞ ) ) which contradicts part (1).(4) For every N ∈ N nf ( H ( ∞ ) , Λ) there is M ∈ M nf ( H ( ∞ ) , Λ) with N ⊇ M .Let S := { M ∈ M nf ( H ( ∞ ) , Λ) | N M } . Then [ M S , N ] ⊆ QZ( H ( ∞ ) ) as above.On the other hand, for T := M nf ( H ( ∞ ) , Λ) , the group M T ⊆ H ( ∞ ) is closed,non-discrete and normal in H , thus M T = H ( ∞ ) . Using (1), we conclude that S = T which proves the assertion.(5) Let S, S ′ be disjoint subsets of M nf ( H ( ∞ ) , Λ) . Then M S ∩ M S ′ ⊆ QZ( H ( ∞ ) ) .If not, we have M S ∩ M S ′ ∈ M nf ( H ( ∞ ) , Λ) and there is, by part (4), an ele-ment M ∈ M nf ( H ( ∞ ) , Λ) with M ⊆ M S ∩ M S ′ . However, this implies that [ M, M ] ⊆ [ M S , M S ′ ] ⊆ QZ( H ( ∞ ) ) which contradicts part (3).(6) The set M nf ( H ( ∞ ) , Λ) is finite.Let G = S M S , where the union is taken over all finite subsets S of the set M nf ( H ( ∞ ) , Λ) . Then G is non-discrete and normal in H . Hence G = H ( ∞ ) by Proposition 2.11 (iii). Since H is second-countable and locally compact,it is metrizable. Hence H ( ∞ ) is a separable metric space and the same holdsfor G . Let L ⊆ G be a countable dense subgroup, and fix an exhaustion F ⊆ F ⊆ · · · ⊆ F of F by finite sets. Let ( S n ) n ∈ N be an increasing sequenceof finite subsets of M nf ( H ( ∞ ) , Λ) such that F n ⊆ M S n . In particular L ⊆ M S n ∈ N S n and thus M S n ∈ N S n = H ( ∞ ) which by (5) and (1) implies M nf ( H ( ∞ ) , Λ) = S n ∈ N S n . Thus M nf ( H ( ∞ ) , Λ) is countable. Next, fix M ∈ M nf ( H ( ∞ ) , Λ) . Then N H ( M ) is closed and ofcountable index in H , and thus has non-empty interior as H is a Baire space.Hence N H ( M ) is open in H . Given that N H ( M ) contains H ( ∞ ) we concludethat N H ( M ) is of finite index in H using Proposition 2.11 (i). Since H actstransitively by on M nf ( H ( ∞ ) , Λ) by (2) we conclude that M nf ( H ( ∞ ) , Λ) isfinite by the orbit-stabilizer theorem. The above claims yield parts (i)(a), (i)(b), (ii)(a) and (iii) of Proposition 2.12.We now turn to parts (ii)(b) and (ii)(c).(ii)(b) Using part (6), let M nf ( H ( ∞ ) , Λ) = { M , . . . , M r } and define Ω := QZ( M ) · . . . · QZ( M r ) . Note that since
QZ( M i ) is characteristic in M i , which is normal in H ( ∞ ) , thequasi-centers in the above definition normalize each other, so Ω is a group.It is then normal in H . If Ω does not act freely on E then Ω \ Γ is finiteby Lemma 2.9 and there exist λ , . . . , λ k ∈ Ω by Lemma 2.3 such that for Ω ′ := h λ , . . . , λ k i the quotient Ω ′ \ Γ is finite. For every i ∈ { , . . . , k } , write λ i = a i b i where a i ∈ QZ( M ) and b i ∈ QZ( M ) · . . . · QZ( M r ) . Let U ≤ M be an open subgroup such that [ a i , U ] = { e } for all i ∈ { , . . . , k } . Since [ M · . . . M r , M ] ⊆ QZ( H ( ∞ ) ) , there is an open subgroup U ≤ M suchthat [ b i , U ] = { e } for all i ∈ { , . . . , k } . Hence U := U ∩ U ≤ M iscontained in Z Aut(Γ) (Ω ′ ) which by Lemma 2.4 implies that U and hence M is discrete, a contradiction. Thus Ω acts freely on E , is discrete and therefore Ω ⊆ QZ( H ( ∞ ) ) . That is QZ( M i ) ⊆ QZ( H ( ∞ ) ) ∩ M i . The opposite inclusionfollows from the definitions.(ii)(c) Let M ∈ M nf ( H ( ∞ ) , Λ) and N E M a closed subgroup containing QZ( M ) .For every M ′ ∈ M nf ( H ( ∞ ) , Λ) with M = M ′ we have [ M ′ , M ] ⊆ M ′ ⊆ M ⊆ QZ( H ( ∞ ) ) This implies [ M ′ , N ] ⊆ QZ( H ( ∞ ) ) ∩ M = QZ( M ) ⊆ N , i.e. M ′ normalizes N .Since N E M , this implies N E H ( ∞ ) and hence, by minimality of M , we haveeither N = M or N acts freely on E and N ⊆ QZ( H ( ∞ ) ) ∩ M = QZ( M ) . (cid:3) Corollary 2.13.
Let
Γ = (
V, E ) be a locally finite, connected graph. Further, let H ≤ Aut(Γ) be closed, non-discrete and locally semiprimitive. Minimal, non-trivialclosed normal subgroups of H ( ∞ ) / QZ( H ( ∞ ) ) exist. They are all H -conjugate, finitein number and topologically simple. Proof.
Apply Proposition 2.12 to
Λ = QZ( H ( ∞ ) ) . (cid:3) We summarize the previous results in the following theorem, which is a verbatimcopy of Burger–Mozes’ Theorem 1.2, except that the local action need only besemiprimitive, not quasiprimitive.
Theorem 2.14.
Let Γ be a locally finite, connected graph. Further, let H ≤ Aut(Γ) be closed, non-discrete and locally semiprimitive. Then(i) H ( ∞ ) is minimal closed normal cocompact in H ,(ii) QZ( H ) is maximal discrete normal, and non-cocompact in H , and(iii) H ( ∞ ) / QZ( H ( ∞ ) ) = H ( ∞ ) / (QZ( H ) ∩ H ( ∞ ) ) admits minimal, non-trivial closednormal subgroups; finite in number, H -conjugate and topologically simple.If Γ is a tree, and, in addition, H is locally primitive then(iv) H ( ∞ ) / QZ( H ( ∞ ) ) is a direct product of topologically simple groups. Proof.
Parts (i) and (ii) stem from parts (i), (ii) and (iii) of Proposition 2.11 in com-bination with Section 1.2. For part (iii), use part (iv) of Proposition 2.11 and Corol-lary 2.13. Finally, part (iv) is Corollary 1.7.2 in [BM00]. It follows from Theorem1.7.1 in [BM00] as the commutator of any two distinct elements in M nf ( H ( ∞ ) , Λ) is contained in QZ( H ( ∞ ) ) . (cid:3) ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 13 Universal Groups
In this section, we develop a generalization of Burger–Mozes universal groupsthat arises through prescribing the local action on balls of a given radius k ∈ N around vertices. The Burger–Mozes construction corresponds to the case k = 1 .Whereas many properties of the original construction carry over to the newsetup, others require adjustments. Notably, there are compatibility and discretenessconditions on the local action F under which the associated universal group islocally action isomorphic to F and discrete respectively.We then exhibit examples and (non)-rigidity phenomena of our construction.Finally, a universality statement holds under an additional assumption.3.1. Definition and Basic Properties.
Definition.
Let Ω be a set of cardinality d ∈ N ≥ and let T d = ( V, E ) denotethe d -regular tree. A labelling l of T d is a map l : E → Ω such that for every x ∈ V the map l x : E ( x ) → Ω , y l ( y ) is a bijection, and l ( e ) = l ( e ) for all e ∈ E .For every k ∈ N , fix a tree B d,k which is isomorphic to a ball of radius k arounda vertex in T d . Let b denote its center and carry over the labelling of T d to B d,k viathe chosen isomorphism. Then for every x ∈ V there is a unique, label-respectingisomorphism l kx : B ( x, k ) → B d,k . We define the k -local action σ k ( g, x ) ∈ Aut( B d,k ) of an automorphism g ∈ Aut( T d ) at a vertex x ∈ V via σ k : Aut( T d ) × V → Aut( B d,k ) , ( g, x ) σ k ( g, x ) := l kgx ◦ g ◦ ( l kx ) − . Definition 3.1.
Let F ≤ Aut( B d,k ) and l be a labelling of T d . Define U ( l ) k ( F ) := { g ∈ Aut( T d ) | ∀ x ∈ V : σ k ( g, x ) ∈ F } . The following lemma states that the maps σ k satisfy a cocycle identity whichimplies that U ( l ) k ( F ) is a subgroup of Aut( T d ) for every F ≤ Aut( B d,k ) . Lemma 3.2.
Let x ∈ V and g, h ∈ Aut( T d ) . Then σ k ( gh, x ) = σ k ( g, hx ) σ k ( h, x ) . Proof.
We compute σ k ( gh, x ) = l k ( gh ) x ◦ gh ◦ ( l kx ) − = l k ( gh ) x ◦ g ◦ h ◦ ( l kx ) − == l k ( gh ) x ◦ g ◦ ( l khx ) − ◦ l khx ◦ h ◦ ( l kx ) − = σ k ( g, hx ) σ k ( h, x ) . (cid:3) Basic Properties.
Note that the group U ( l )1 ( F ) of Definition 3.1 coincideswith the Burger–Mozes universal group U ( l ) ( F ) introduced in [BM00, Section 3.2]under the natural isomorphism Aut( B d, ) ∼ = Sym(Ω) . Several basic properties ofthe latter group carry over to the generalized setup. First of all, passing betweendifferent labellings of T d amounts to conjugating in Aut( T d ) . Subsequently, we shalltherefore omit the reference to an explicit labelling. Lemma 3.3.
For every quadruple ( l, l ′ , x, x ′ ) of labellings l, l ′ of T d and vertices x, x ′ ∈ V , there is a unique automorphism g ∈ Aut( T d ) with gx = x ′ and l ′ = l ◦ g . Proof.
Set gx := x ′ . Now assume inductively that g is uniquely determined on B ( x, n ) ( n ∈ N ) and let v ∈ S ( x, n ) . Then g is also uniquely determined on E ( v ) by the requirement l ′ = l ◦ g , namely g | E ( v ) := l | − E ( gv ) ◦ l ′ | E ( v ) . (cid:3) Proposition 3.4.
Let F ≤ Aut( B d,k ) . Further, let l and l ′ be labellings of T d .Then the groups U ( l ) k ( F ) and U ( l ′ ) k ( F ) are conjugate in Aut( T d ) . Proof.
Choose x ∈ V . Let τ ∈ Aut( T d ) denote the automorphism of T d associatedto ( l, l ′ , x, x ) by Lemma 3.3, then U ( l ) k ( F ) = τ U ( l ′ ) k ( F ) τ − . (cid:3) The following basic properties of U k ( F ) are as in Proposition 1.4. Proposition 3.5.
Let F ≤ Aut( B d,k ) . The group U k ( F ) is(i) closed in Aut( B d,k ) ,(ii) vertex-transitive, and(iii) compactly generated. Proof.
As to (i), note that if g / ∈ U k ( F ) then σ k ( g, x ) / ∈ F for some x ∈ V . In thiscase, the open neighbourhood { h ∈ Aut( T d ) | h | B ( x,k ) = g | B ( x,k ) } of g in Aut( T d ) is also contained in the complement of U k ( F ) .For (ii), let x, x ′ ∈ V and let g ∈ Aut( T d ) be the automorphism of T d associatedto ( l, l, x, x ′ ) by Lemma 3.3. Then g ∈ U k ( F ) as σ k ( g, v ) = id ∈ F for all v ∈ V .To prove (iii), fix x ∈ V . We show that U k ( F ) is generated by the join of thecompact set U k ( F ) x and the finite generating set of U ( { id } ) = U k ( { id } ) ≤ U k ( F ) guaranteed by Lemma 1.5: Indeed, for g ∈ U k ( F ) pick g ′ in the finitely generated,vertex-transitive subgroup U ( { id } ) of U k ( F ) such that g ′ gx = x . We then have g ′ g ∈ U k ( F ) x and the assertion follows. (cid:3) For completeness, we explicitly state the following.
Proposition 3.6.
Let F ≤ Aut( B d,k ) . Then U k ( F ) is a compactly generated,totally disconnected, locally compact, second countable group. Proof.
The group U k ( F ) is totally disconnected, locally compact, second countableas a closed subgroup of Aut( T d ) and compactly generated by Proposition 3.5. (cid:3) Finally, we record that the groups U k ( F ) are k -closed. Proposition 3.7.
Let F ≤ Aut( B d,k ) . Then U k ( F ) satisfies Property P k . Proof.
Let e = ( x, y ) ∈ E . Clearly, U k ( F ) e k ⊇ U k ( F ) e k ,T y · U k ( F ) e k ,T x . Conversely,consider g ∈ U k ( F ) e k and define g y ∈ Aut( T d ) and g x ∈ Aut( T d ) by σ k ( g y , v ) = ( σ k ( g, v ) v ∈ V ( T x )id v ∈ V ( T y ) and σ k ( g x , v ) = ( id v ∈ V ( T x ) σ k ( g, v ) v ∈ V ( T y ) respectively. Then g y ∈ U k ( F ) e k ,T y , g x ∈ U k ( F ) e k ,T x and g = g y ◦ g x . (cid:3) Compatibility and Discreteness.
We now generalize parts (iv) and (vi) ofBurger–Mozes’ Proposition 1.4. There are compatibility and discreteness conditions(C) and (D) on subgroups F ≤ Aut( B d,k ) that hold if and only if the associateduniversal group is locally action isomorphic to F and discrete respectively.We introduce the following notation for vertices in the labelled tree ( T d , l ) : Given x ∈ V and w = ( ω , . . . , ω n ) ∈ Ω n ( n ∈ N ) , set x w := γ x,w ( n ) where γ x,w : Path ( w ) n := b b b b . . . nω ω → T d is the unique label-respecting morphism sending to x ∈ V . If w is the emptyword, set x w := x . Whenever admissible, we also adopt this notation in the caseof B d,k and its labelling. In particular, S ( x, n ) is in natural bijection with the set Ω ( n ) := { ( ω , . . . , ω n ) ∈ Ω n | ∀ k ∈ { , . . . , n − } : ω k = ω k +1 } .3.2.1. Compatibility.
First, we ask whether U k ( F ) locally acts like F , that is whetherthe actions U k ( F ) x y B ( x, k ) and F y B d,k are isomorphic for every x ∈ V .Whereas this always holds for k = 1 by Proposition 1.4(iv) it need not be true for k ≥ , the issue being (non)-compatibility among elements of F . See Example 3.9.The condition developed in this section allows for computations. A more practicalversion from a theoretical viewpoint follows in Section 3.4. ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 15
Now, let x ∈ V and suppose that α ∈ U k ( F ) x realizes a ∈ F at x , that is α | B ( x,k ) = ( l kx ) − ◦ a ◦ l kx . Then given the condition that σ k ( α, x ω ) be in F for all ω ∈ Ω , we obtain thefollowing necessary compatibility condition on F for U k ( F ) to act like F at x ∈ V : ∀ a ∈ F ∀ ω ∈ Ω : ∃ a ω ∈ F : ( l kx ) − ◦ a ◦ l kx | S ω = ( l kαx ω ) − ◦ a ω ◦ l kx ω | S ω where S ω := B ( x, k ) ∩ B ( x ω , k ) ⊆ T d . Set T ω := l kx ( S ω ) ⊆ B d,k . Then the abovecondition can be rewritten as ∀ a ∈ F ∀ ω ∈ Ω : ∃ a ω ∈ F : a ω | T ω = l kαx ω ◦ ( l kx ) − ◦ a ◦ l kx ◦ ( l kx ω ) − | T ω . Now observe the following: First, αx ω depends only on a . Second, the subtree T ω of B d,k does not depend on x . Third, ι ω := l kx | T ω ◦ ( l kx ω ) − | T ω is the unique non-trivial,involutive and label-respecting automorphism of T ω ; it is given by ι ω := l kx (cid:12)(cid:12) T ω ◦ ( l kx ω ) − (cid:12)(cid:12) T ω : T ω → S ω → T ω , b w x ωw b ωw for admissible words w . Hence the above condition may be rewritten as(C) ∀ a ∈ F ∀ ω ∈ Ω : ∃ a ω ∈ F : a ω | T ω = ι aω ◦ a ◦ ι ω . In this situation we shall say that a ω is compatible with a in direction ω . Proposition 3.8.
Let F ≤ Aut( B d,k ) . Then U k ( F ) is locally action isomorphic to F if and only if F satisfies (C). Proof.
By the above, condition (C) is necessary. To show that it is also sufficient,let x ∈ V and a ∈ F . We aim to define an automorphism α ∈ U k ( F ) which realizes a at x . This forces us to define α | B ( x,k ) := ( l kx ) − ◦ a ◦ l kx . Now, assume inductively that α is defined consistently on B ( x, n ) in the sense that σ k ( α, y ) ∈ F for all y ∈ B ( x, n ) with B ( y, k ) ⊆ B ( x, n ) . In order to extend α to B ( x, n + 1) , let y ∈ S ( x, n − k + 1) and let ω ∈ Ω be the unique label such that y ω ∈ S ( x, n − k ) . Set c := σ k ( α, y ω ) . Applying condition (C) to the pair ( c, ω ) yieldsan element c ω ∈ F such that ( l kαy ω ) − ◦ c ◦ l ky ω (cid:12)(cid:12) S ω = ( l kαy ) − ◦ c ω ◦ l ky (cid:12)(cid:12) S ω where S ω := B ( y, k ) ∩ B ( y ω , k ) and we have realized ι ω as l ky ω (cid:12)(cid:12) T ω ◦ ( l ky ) − (cid:12)(cid:12) T ω and ι cω as l kαy (cid:12)(cid:12) T cω ◦ ( l kαy ω ) − (cid:12)(cid:12) T cω . Now extend α consistently to B ( v, n + 1) by setting α | B ( x,k ) := ( l kαx ) − ◦ c ω ◦ l kx . (cid:3) Example 3.9.
Let
Ω := { , , } and a ∈ Aut( B , ) be the element which swapsthe leaves b and b of B , . Then F := h a i = { id , a } does not contain an elementcompatible with a in direction ∈ Ω and hence does not satisfy condition (C).We show that it suffices to check condition (C) on the elements of a generating set.Let F ≤ Aut( B d,k ) and a, b ∈ F . Set c := ab . Then c ω | T ω = ι cω ◦ a ◦ b ◦ ι ω = ( ι cω ◦ a ◦ ι bω ) ◦ ( ι bω ◦ b ◦ ι ω )= (cid:0) ι a ( bω ) ◦ a ◦ σ bω (cid:1) ◦ ( ι bω ◦ b ◦ ι ω ) . (M)Let C F ( a, ω ) denote the compatibility set of elements in F which are compatible with a ∈ F in direction ω ∈ Ω . Then (M) shows that C F ( ab, ω ) ⊇ C F ( a, bω ) C F ( b, ω ) . Ittherefore suffices to check condition (C) on a generating set of F .Given S ⊆ Ω , we also define C F ( a, S ) := T ω ∈ S C F ( a, ω ) , the set of elements in F which are compatible with a ∈ F in all directions from S . We omit F in thisnotation when it is clear from the context. As a consequence, we obtain the following description of the local action of U k ( F ) when F does not satisfy condition (C). Proposition 3.10.
Let F ≤ Aut( B d,k ) . Then F has a unique maximal subgroup C ( F ) which satisfies (C). We have C ( C ( F )) = C ( F ) and U k ( F ) = U k ( C ( F )) . Proof.
By the above, C ( F ) := h H ≤ F | H satisfies (C) i ≤ F satisfies condition (C).It is the unique maximal such subgroup of F by definition, and C ( C ( F )) = C ( F ) .Furthermore, U k ( C ( F )) ≤ U k ( F ) . Conversely, suppose g ∈ U k ( F ) \ U k ( C ( F )) .Then there is x ∈ V such that σ k ( g, x ) ∈ F \ C ( F ) and the group C ( F ) (cid:12) h C ( F ) , { σ k ( g, x ) | x ∈ V }i ≤ F satisfies condition (C), too, as can be seen by setting σ k ( g, x ) ω := σ k ( g, x ω ) . Thiscontradicts the maximality of C ( F ) . (cid:3) Remark 3.11.
Let F ≤ Aut( B d,k ) satisfy (C). The proof of Proposition 3.8 showsthat elements of U k ( F ) are readily constructed: Given x, y ∈ V ( T d ) and a ∈ F ,define g : B ( x, k ) → B ( y, k ) by setting g ( x ) = y and σ k ( g, x ) = a . Then, givenelements a ω ∈ F ( ω ∈ Ω) such that a ω ∈ C ( a, ω ) for all ω ∈ Ω , there is a uniqueextension of g to B ( x, k +1) so that σ k ( g, x ω ) = a ω for all ω ∈ Ω . Proceed iteratively.3.2.2. Discreteness.
The group F ≤ Aut( B d,k ) also determines whether or not U k ( F ) is discrete. In fact, the following proposition generalizes Proposition 1.4(vi). Proposition 3.12.
Let F ≤ Aut( B d,k ) . Then U k ( F ) is discrete if F satisfies(D) ∀ ω ∈ Ω : F T ω = { id } . Conversely, if U k ( F ) is discrete and F satisfies (C), then F satisfies (D).Alternatively, U k ( F ) is discrete if and only if C ( F ) satisfies (D). Example 3.9shows that condition (C) is necessary for the second part of Proposition 3.12.Finally, note that F satisfies (D) if and only if C F (id , ω ) = { id } for all ω ∈ Ω . Proof. (Proposition 3.12). Fix x ∈ V . A subgroup H ≤ Aut( T d ) is non-discrete ifand only if for every n ∈ N there is h ∈ H \{ id } such that h | B ( x,n ) = id .Suppose that U k ( F ) is non-discrete. Then there are n ∈ N ≥ k and α ∈ U k ( F ) such that α | B ( x,n ) = id and α | B ( x,n +1) = id . Hence there is y ∈ S ( x, n − k + 1) with a := σ k ( α, y ) = id . In particular, a ∈ F T ω \{ id } where ω is the label of the uniqueedge e ∈ E with o ( e ) = y and d ( x, y ) = d ( x, t ( e )) + 1 .Conversely, suppose that F satisfies (C) and F T ω = { id } for some ω ∈ Ω . Thenfor every n ∈ N ≥ k , we define an automorphism α ∈ U k ( F ) with α | B ( x,n ) = id and α | B ( x,n +1) = id : If α | B ( x,n ) = id , then σ k ( α, y ) ∈ F for all y ∈ B ( x, n − k ) . Choose e ∈ E with y := o ( e ) ∈ S ( x, n − k + 1) and t ( e ) ∈ S ( x, n − k ) such that l ( e ) = ω .We extend α to B ( y, k ) by setting α | B ( y,k ) := l ky ◦ s ◦ ( l ky ) − where s ∈ F T ω \{ id } .Finally, we extend α to T d using (C). (cid:3) We define condition (CD) on F ≤ Aut( B d,k ) as the conjunction of (C) and (D).The following description is immediate from the above.(CD) ∀ a ∈ F ∀ ω ∈ Ω : ∃ ! a ω ∈ F : a ω | T ω = ι aω ◦ a ◦ ι ω . When F satisfies (CD), an element of U k ( F ) x is determined by its action on B ( x, k ) .Hence U k ( F ) x ∼ = F for every x ∈ V and U k ( F ) ( x,y ) ∼ = F ( b,b ω ) for every ( x, y ) ∈ E with l ( x, y ) = ω . Furthermore, F admits a unique involutive compatibility cocycle ,i.e. a map z : F × Ω → F, ( a, ω ) a ω which for all a, b ∈ F and ω ∈ Ω satisfies(i) (compatibility) z ( a, ω ) ∈ C F ( a, ω ) ,(ii) (cocycle) z ( ab, ω ) = z ( a, bω ) z ( b, ω ) , and(iii) (involutive) z ( z ( a, ω ) , ω ) = a .Note that z restricts to an automorphism z ω of F ( b,b ω ) ( ω ∈ Ω) of order at most . ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 17
Group Structure.
For e F ≤ Aut( B d,k ) , let F := π e F ≤ Sym(Ω) denote theprojection of e F onto Aut( B d, ) ∼ = Sym(Ω) . As an illustration, we record that thegroup structure of U k ( e F ) is particularly simple if F is regular. Proposition 3.13.
Let e F ≤ Aut( B d,k ) satisfy (C). Suppose F := π e F is regular.Then U k ( e F ) = U ( F ) ∼ = F ∗ Z / Z . Proof.
Fix x ∈ V . Since F is transitive, the group U k ( e F ) is generated by U k ( e F ) x and an involution ι inverting an edge with origin x . Given α ∈ U k ( e F ) x , regularityof F implies that σ ( α, y ) = σ ( α, x ) ∈ F for all y ∈ V . Now, the subgroups H := U k ( e F ) x ∼ = F and H := h ι i of U k ( e F ) generate a free product within U k ( F ) by the ping-pong lemma: Put X := V ( T x ) and X := V ( T x ω ) . Any non-trivialelement of H maps X into X as F ω = { id } , and ι ∈ H maps X into X . (cid:3) More generally, Bass-Serre theory [Ser03] identifies the universal groups U k ( F ) as amalgamated free products. Proposition 3.14.
Let F ≤ Aut( B d,k ) satisfy (C) (and (D)). If πF is transitive, U k ( F ) ∼ = U k ( F ) x ∗ U k ( F ) ( x,y ) U k ( F ) { x,y } ∼ = F ∗ F ( b,bω ) ( F ( b,b ω ) ⋊ Z / Z ) ! for any edge ( x, y ) ∈ E , where ω = l ( x, y ) and Z / Z acts on F ( b,b ω ) as z ω . Corollary 3.15.
Let
F, F ′ ≤ Aut( B d,k ) satisfy (CD). If there are ω, ω ′ ∈ Ω and anisomorphism ϕ : F → F ′ such that ϕ ( F ( b,b ω ) ) = F ′ ( b,b ω ′ ) , then U k ( F ) ∼ = U k ( F ′ ) . (cid:3) Note that Corollary 3.15 applies to conjugate subgroups of
Aut( B d,k ) whichsatisfy (CD). The following example shows that the assumption that both F and F ′ in Corollary 3.15 satisfy (CD) is indeed necessary. Example 3.16.
Let
Ω := { , , } and t ∈ Aut( B , ) be the element which swapsthe leaves x and x of B , . Using the notation of Section 3.4.1, consider thegroup Γ( A ) ≤ Aut( B , ) which satisfies (C). In particular, U (Γ( A )) ∼ = A ∗ Z / Z by Proposition 3.13. On the other hand, set F ′ := t Γ( A ) t − . Then πF ′ = A while for a non-trivial element α of F ′ , we have σ ( α, b ω ) ∈ S \ A for some ω ∈ Ω .Therefore, U ( F ′ ) = U ( { id } ) is isomorphic to Z / Z ∗ Z / Z ∗ Z / Z by Lemma 1.5.In particular, U (Γ( A )) and U ( t Γ( A ) t − ) are not isomorphic.Conversely, the following Proposition based on [Rad17, Appendix A], whichstates that in certain cases the tree can be recovered from the topological groupstructure of a subgroup of Aut( T d ) , applies to appropriate universal groups. Proposition 3.17.
Let
H, H ′ ≤ Aut( T d ) be closed and locally transitive withdistinct point stabilizers. Then H and H ′ are isomorphic topological groups if andonly if they are conjugate in Aut( T d ) . Proof.
By [FTN91], every compact subgroup of H is either contained in a vertexstabilizer H x ( x ∈ V ) or, in case H Aut( T d ) + , in a geometric edge stabilizer H { e,e } ( e ∈ E ) . Since H is locally transitive, the above are pairwise distinct.The vertex stabilizers are precisely those maximal compact subgroups K ≤ H for which there is no maximal compact subgroup K ′ with [ K : K ∩ K ′ ] = 2 :Indeed, for e ∈ E and x ∈ { o ( e ) , t ( e ) } we have [ H { e,e } : H { e,e } ∩ H x ] = 2 whereas [ H x : H x ∩ H y ] , [ H x : H x ∩ H { e,e } ] ≥ for all distinct x, y ∈ V and e ∈ E by theorbit-stabilizer theorem because d ≥ and H is locally transitive.Adjacency can be expressed in terms of indices as well: Let x, y ∈ V be distinct.Then ( x, y ) ∈ E if and only if [ H x : H x ∩ H y ] ≤ [ H x : H x ∩ H z ] for all z ∈ V :Indeed, if ( x, y ) ∈ E , then [ H x : H x ∩ H y ] = d by the orbit-stabilizer theorem given that H is locally transitive. If z ∈ V is not adjacent to x then [ H x : H x ∩ H z ] > d because point stabilizers of every local action of H are distinct.Now, let Φ : H → H ′ be an isomorphism of topological groups. Then Φ inducesa bijection between the maximal compact subgroups of H and H ′ , and preservesindices. Hence there is an automorphism ϕ ∈ Aut( T d ) such that Φ( H x ) = H ′ ϕ ( x ) forall x ∈ V . Furthermore, since vertex stabilizers in H ′ are pairwise distinct and H ′ ϕhϕ − ( x ) = Φ( H hϕ − ( x ) ) = Φ( hH ϕ − ( x ) h − ) = Φ( h ) H ′ x Φ( h − ) = H ′ Φ( h ) x for all x ∈ V we have ϕhϕ − = Φ( h ) for all h ∈ H . (cid:3) The following Corollary uses the notation Φ k ( F ′ ) from Section 3.4.2. Corollary 3.18.
Let F ≤ Aut( B d,k ) and F ′ ≤ Aut( B d,k ′ ) satisfy (C). Assume k ≥ k ′ and πF, πF ′ ≤ Sym(Ω) are transitive with distinct point stabilizers. If U k ( F ) and U k ′ ( F ′ ) are isomorphic topological groups then F, Φ k ( F ′ ) ≤ Aut( B d,k ) are conjugate. Proof.
By Proposition 3.17, the groups U k ( F ) and U k ( F ′ ) are conjugate in Aut( T d ) ,hence so are U k ( F ) x and U k ′ ( F ′ ) x for every x ∈ V and the assertion follows. (cid:3) Example 3.19.
Section 3.4.1 introduces the isomorphic, non-conjugate subgroups Π( S , sgn , { } ) and Π( S , sgn , { , } ) of Aut( B , ) , both of which project onto S and satisfy (C) but not (D). An explicit isomorphism satisfies the assumption ofCorollary 3.15. However, by Corollary 3.18 the universal groups U (Π( S , sgn , { } )) and U (Π( S , sgn , { , } )) are non-isomorphic. Therefore, Corollary 3.15 does notgeneralize to the non-discrete case. Question 3.20.
Let
F, F ′ ≤ Aut( B d,k ) satisfy (C) and be conjugate. Are theassociated universal groups U k ( F ) and U k ( F ′ ) necessarily isomorphic?In the following, we determine the Burger–Mozes subquotient H ( ∞ ) / QZ( H ( ∞ ) ) of Theorem 2.14 for non-discrete, locally semiprimitive universal groups. Proposition 3.21.
Let F ≤ Aut( B d,k ) satisfy (C). If, in addition, F satisfies (D)then QZ(U k ( F )) = U k ( F ) . Otherwise, QZ(U k ( F )) = { id } . Proof. If F satisfies (D) then U k ( F ) is discrete and hence QZ(U k ( F )) = U k ( F ) .Conversely, if F satisfies (C) but not (D) then the stabilizer of any half-tree T ⊆ T d in U k ( F ) is non-trivial: We have T ∈ { T x , T y } for some edge e := ( x, y ) ∈ E . Since U k ( F ) is non-discrete by Proposition 3.12 and satisfies Property P k by Proposition3.7, the group U k ( F ) e k = U k ( F ) e k ,T y · U k ( F ) e k ,T x is non-trivial. In particular, either U k ( F ) T x or U k ( F ) T y is non-trivial. In view of the existence of label-respectinginversions, both are non-trivial and hence so is U k ( F ) T . Therefore, U k ( F ) hasProperty H of Möller–Vonk [MV12, Definition 2.3] and [MV12, Proposition 2.6]implies that U k ( F ) has trivial quasi-center. (cid:3) Proposition 3.22.
Let F ≤ Aut( B d,k ) satisfy (C) but not (D). Suppose that πF is semiprimitive. Then U k ( F ) ( ∞ ) / QZ(U k ( F ) ( ∞ ) ) = U k ( F ) ( ∞ ) = U k ( F ) + k . Proof.
The subgroup U k ( F ) + k ≤ U k ( F ) is open, hence closed, and normal in U k ( F ) by definition. Since U k ( F ) is non-discrete by Proposition 3.12, so is U k ( F ) + k . UsingProposition 2.11(iii), we conclude that U k ( F ) + k ≥ U k ( F ) ( ∞ ) . Since U k ( F ) satisfiesProperty P k by Proposition 3.7, the group U k ( F ) + k is simple due to Theorem 1.1.Thus U k ( F ) + k = U k ( F ) ( ∞ ) . Given that QZ(U k ( F ) ( ∞ ) ) = QZ(U k ( F )) ∩ U k ( F ) ( ∞ ) by Proposition 2.11(iv), the assertion follows from Proposition 3.21. (cid:3) In the context of Proposition 3.22, the group U k ( F ) + k is simple, compactlygenerated, non-discrete, totally disconnected, locally compact, second countable.Compact generation follows from [KM08, Corollary 2.11] given that U k ( F ) + k iscocompact in U k ( F ) by Proposition 2.11(i). ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 19
Examples.
We now construct various classes of examples of subgroups of
Aut( B d,k ) satisfying (C) or (CD), and prove a rigidity result for certain local actions.First, we give a suitable realization of Aut( B d,k ) and the conditions (C) and (D).Namely, we view an automorphism α of B d,k as the set { σ k − ( α, v ) | v ∈ B ( b, } as follows: Let Aut( B d, ) ∼ = Sym(Ω) be the natural isomorphism. For k ≥ , weiteratively identify Aut( B d,k ) with its image under the map Aut( B d,k ) → Aut( B d,k − ) ⋉ Y ω ∈ Ω Aut( B d,k − ) , α ( σ k − ( α, b ) , ( σ k − ( α, b ω )) ω ) where Aut( B d,k − ) acts on Q ω ∈ Ω Aut( B d,k − ) by permuting the factors accordingto its action on S ( b, ∼ = Ω . That is, multiplication in Aut( B d,k ) is given by ( α, ( α ω ) ω ∈ Ω ) ◦ ( β, ( β ω ) ω ∈ Ω ) = ( αβ, ( α βω β ω ) ω ∈ Ω ) . Consider the homomorphism π k − : Aut( B d,k ) → Aut( B d,k − ) , α σ k − ( α, b ) ,the projections pr ω : Aut( B d,k ) → Aut( B d,k − ) , α σ k − ( α, b ω ) ( ω ∈ Ω) , and p ω = ( π k − , pr ω ) : Aut( B d,k ) → Aut( B d,k − ) × Aut( B d,k − ) , whose image we interpret as a relation on Aut( B d,k − ) . The conditions (C) and (D)for a subgroup F ≤ Aut( B d,k ) now read as follows.(C) ∀ ω ∈ Ω : p ω ( F ) is symmetric(D) ∀ ω ∈ Ω : p ω | − F (id , id) = { id } The case k = 2 . We first consider the case k = 2 which is all-encompassingin certain situations, see Theorem 3.32.Consider the map γ : Sym(Ω) → Aut( B d, ) , a ( a, ( a, . . . , a )) ∈ Aut( B d, ) ,using the realization of Aut( B d, ) from above. For every F ≤ Sym(Ω) , the image Γ( F ) := im( γ | F ) = { ( a, ( a, . . . , a )) | a ∈ F } ∼ = F is a subgroup of Aut( B d, ) which is isomorphic to F and satisfies both (C) and (D).The involutive compatibility cocycle is given by Γ( F ) × Ω → Γ( F ) , ( γ ( a ) , ω ) γ ( a ) .Note that Γ( F ) implements the diagonal action F y Ω on Ω (2) ∼ = S ( b, .We obtain U (Γ( F )) = { α ∈ Aut( T d ) | ∃ a ∈ F : ∀ x ∈ V : σ ( α, x ) = a } =: D( F ) ,following the notation of [BEW15]. Moreover, there is the following description ofall subgroups e F ≤ Aut( B d, ) with π e F = F that satisfy (C) and contain Γ( F ) . Proposition 3.23.
Let F ≤ Sym(Ω) . Given K ≤ Q ω ∈ Ω F ω ∼ = ker π ≤ Aut( B d, ) ,there is e F ≤ Aut( B d, ) satisfying (C) and fitting into the split exact sequence / / K / / ι / / e F π / F γ o / / if and only if K is preserved by the action F y Q ω ∈ Ω F ω , a · ( a ω ) ω := ( aa a − ω a − ) ω . Proof.
If there is a split exact sequence as above then K E e F is invariant underconjugation by Γ( F ) ≤ e F , hence the assertion.Conversely, if K is invariant under the given action, then e F := { ( a, ( aa ω ) ω ) | a ∈ F, ( a ω ) ω ∈ K } fits into the sequence: First, note that e F contains both K and Γ( F ) . It is also asubgroup of Aut( B d, ) : For ( a, ( aa ω ) ω ) , ( b, ( bb ω ) ω ) ∈ e F we have ( a, ( aa ω ) ω ) ◦ ( b, ( bb ω ) ω ) = ( ab, ( aa bω bb ω ) ω ) = ( ab, ( ab ◦ b − a bω b ◦ b ω ) ω ) ∈ e F by assumption. In particular, e F = h Γ( F ) , K i . It suffices to check condition (C) onthese generators of e F . As before, γ ( a ) ∈ C ( γ ( a ) , ω ) for all a ∈ F and ω ∈ Ω . Nowlet k ∈ K . Then γ (pr ω k ) k − ∈ C ( k, ω ) for all ω ∈ Ω . (cid:3) Example 3.24.
We show that for certain dihedral groups there are only fourgroups of the type given in Proposition 3.23: Set F := D p ≤ Sym( p ) for someprime p ≥ . Then F ω ∼ = ( F , +) . Hence U := Q ω ∈ Ω F ω is a p -dimensional vectorspace over F and the F -action on it permutes coordinates. When ∈ ( Z /p Z ) ∗ is primitive, there are only four F -invariant subspaces of U : The trivial subspace,the diagonal subspace h (1 , . . . , i , the whole space, and K := ker σ ∼ = F ( p − where σ : U → F is given by ( v , . . . , v p ) P pi =1 v i . Note that K is F -invariant becausethe homomorphism σ is. Conjecturally, there are infinitely many primes for which ∈ ( Z /p Z ) ∗ is primitive. The list starts with , , , , . . . , see [Slo, A001122].Suppose that W ≤ U is F -invariant. It suffices to show that W contains K assoon as W ∩ ker σ contains a non-trivial element w . To see this, we show that theorbit of w under the cyclic group h ̺ i = C p ≤ D p generates a ( p − -dimensionalsubspace of K which hence equals K : Indeed, the rank of the circulant matrix C := ( w, ̺w, ̺ w, . . . , ̺ ( p − w ) equals p − deg(gcd( x p − , f ( x ))) where f ( x ) ∈ F [ x ] is the polynomial f ( x ) = w p x p − + · · · + w x + w , see e.g. [Day60, Corollary 1]. Thepolynomial x p − ∈ F [ x ] factors into the irreducibles ( x p − + x p − + · · · + x +1)( x − by the assumption on p . Since f has an even number of non-zero coefficients, weconclude that rank( C ) = p − .The following subgroups of Aut( B d, ) are of the type given in Proposition 3.23.Let F ≤ Sym(Ω) be transitive. Fix ω ∈ Ω , let C ≤ Z ( F ω ) and let N E F ω benormal. Furthermore, fix elements f ω ∈ F ( ω ∈ Ω ) satisfying f ω ( ω ) = ω . We define ∆( F, C ) := { ( a, ( a ◦ f ω a f − ω ) ω ) | a ∈ F, a ∈ C } ∼ = F × C, and Φ( F, N ) := { ( a, ( a ◦ f ω a ( ω )0 f − ω ) ω ) | a ∈ F, ∀ ω ∈ Ω : a ( ω )0 ∈ N } ∼ = F ⋉ N d . In the case of ∆( F, C ) we have K = { ( f ω a f − ω ) ω | a ∈ C } whereas in thecase of Φ( F, N ) we have K = { ( f ω a ( ω )0 f − ω ) ω | ∀ ω ∈ Ω : a ( ω )0 ∈ N } . In both cases,invariance under the action of F is readily verified, as is condition (D) for ∆( F, C ) .The group ∆( F, F ω ) can be defined for non-abelian F ω as well, namely ∆( F ) := { ( a, ( f aω f − ω ◦ f ω a f − ω ) ω ) | a ∈ F, a ∈ F ω } ∼ = F × F ω . However, it need not contain Γ( F ) . Note that Φ( F, N ) does not depend on the choiceof the elements ( f ω ) ω ∈ Ω as N is normal in F ω , whereas ∆( F, C ) and ∆( F ) may.However, any group of the form { ( a, ( z ( a, ω ) α ω ( a )) ω ) | a ∈ F, a ∈ F ω } , where z is a compatibility cocycle of F and α ω : F ω → F ω ( ω ∈ Ω) are isomorphisms, whichsatisfies (C) and in which { ( a, ( z ( a, ω )) ω ) | a ∈ F } and { (id , ( α ω ( a )) ω ) | a ∈ F ω } commute, will be referred to as ∆( F ) in view of Corollary 3.15.The group Φ( F, F ω ) can be defined without assuming transitivity of F , namely Φ( F ) := { ( a, ( a ω ) ω ) | a ∈ F, ∀ ω ∈ Ω : a ω ∈ C F ( a, ω ) } ∼ = F ⋉ Y ω ∈ Ω F ω . We conclude that U (Φ( F )) = U ( F ) for every F ≤ Sym(Ω) .When F ≤ Sym(Ω) preserves a partition P : Ω = F i ∈ I Ω i of Ω , we define Φ( F, P ) := { ( a, ( a ω ) ω ) | a ∈ F, a ω ∈ C F ( a, ω ) constant w.r.t. P} ∼ = F ⋉ Y i ∈ I F Ω i . The group Φ( F, P ) satisfies (C) as well and features prominently in Section 4.1.The following kind of -local action generalises the sign construction in [Rad17].Let F ≤ Sym(Ω) and ρ : F ։ A a homomorphism to an abelian group A . Define Π( F, ρ, { } ) := n ( a, ( a ω ) ω ) ∈ Φ( F ) (cid:12)(cid:12)(cid:12) Y ω ∈ Ω ρ ( a ω ) = 1 o , and Π( F, ρ, { , } ) := n ( a, ( a ω ) ω ) ∈ Φ( F ) (cid:12)(cid:12)(cid:12) ρ ( a ) Y ω ∈ Ω ρ ( a ω ) = 1 o . This construction is generalised to k ≥ in Section 3.4.2 where the third entry of Π is a set of radii over which the defining product is taken. ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 21
Proposition 3.25.
Let F ≤ Sym(Ω) and ρ : F ։ A a homomorphism to anabelian group A . Let e F ∈ { Π( F, ρ, { } ) , Π( F, ρ, { , } ) } . If ρ ( F ω ) = A for all ω ∈ Ω then π e F = F and e F satisfies (C). Proof. As C F ( a, ω ) = aF ω , and ρ ( F ω ) = A for all ω ∈ Ω , an element ( a, ( a ω ) ω ) ∈ Φ( F ) can be turned into an element of e F by changing a ω for a single, arbitrary ω ∈ Ω .We conclude that π e F = F and that e F satisfies (C). (cid:3) General case.
We extend some constructions of Section 3.4.1 to arbitrary k .Given F ≤ Aut( B d,k ) satisfying (C), define the subgroup Φ k ( F ) := { ( α, ( α ω ) ω ) | α ∈ F, ∀ ω ∈ Ω : α ω ∈ C F ( α, ω ) } ≤ Aut( B d,k +1 ) . Then Φ k ( F ) inherits condition (C) from F and we obtain U k +1 (Φ k ( F )) = U k ( F ) .Concerning the construction Γ we have the following. Proposition 3.26.
Let F ≤ Aut( B d,k ) satisfy (C). Then there exists a group Γ k ( F ) ≤ Aut( B d,k +1 ) satisfying (CD) such that π k : Γ k ( F ) → F is an isomorphismif and only if F admits an involutive compatibility cocycle z . Proof. If F admits an involutive compatibility cocycle z , define Γ k ( F ) := { ( α, ( z ( α, ω )) ω ) | α ∈ F } ≤ Aut( B d,k +1 ) . Then γ z : F → Γ k ( F ) , α ( α, ( z ( α, ω )) ω ) is an isomorphism and the involutivecompatibility cocycle of Γ k ( F ) is given by e z : ( γ z ( α ) , ω ) γ z ( z ( α, ω )) . Conversely,if a group Γ k ( F ) with the asserted properties exists, set z : ( α, ω ) pr ω π − k α . (cid:3) Let F ≤ Aut( B d,k ) satisfy (C) and let l > k . We set Γ l ( F ) := Γ l − ◦ · · · ◦ Γ k ( F ) for an implicit sequence of involutive compatibility cocycles. Similarly, we define Φ l ( F ) := Φ l − ◦ · · · ◦ Φ k ( F ) . Now, let e F ≤ Aut( B d,k ) . Assume F := π e F ≤ Sym(Ω) preserves a partition P : Ω = F i ∈ I Ω i of Ω . Define the group Φ k ( e F , P ) := { ( α, ( α ω ) ω ) | α ∈ e F , α ω ∈ C e F ( α, ω ) is constant w.r.t. P} . If C e F ( α, Ω i ) is non-empty for all α ∈ e F and i ∈ I then Φ k ( e F , P ) satisfies (C), andif C e F (id , Ω i ) is non-trivial for all i ∈ I then Φ k ( e F , P ) does not satisfy (D).The following statement generalizes Proposition 3.23. Proposition 3.27.
Let F ≤ Aut( B d,k ) satisfy (C). Suppose F admits an involutivecompatibility cocycle z . Given K ≤ Φ k ( F ) ∩ ker( π k ) , there is e F ≤ Aut( B d,k +1 ) satisfying (C) and fitting into the split exact sequence / / K / / ι / / e F π / F γ z o / / if and only if Γ k ( F ) normalizes K , and for all k ∈ K and ω ∈ Ω there is k ω ∈ K such that pr ω k ω = z (pr ω k, ω ) − . Proof.
If there is a split exact sequence as above then K E e F is invariant underconjugation by Γ k ( F ) . Moreover, all elements of e F have the form ( α, ( z ( α, ω ) α ω ) ω ) for some α ∈ F and ( α ω ) ω ∈ K . This implies the second assertion on K .Conversely, if K satisfies the assumptions, then e F := { ( α, ( z ( α, ω ) α ω ) ω ) | α ∈ F, ( α ω ) ω ∈ K } fits into the sequence: First, note that e F contains both K and Γ k ( F ) . It is also asubgroup of Aut( B d,k +1 ) : For ( α, ( z ( α, ω ) α ω ) ω ) , ( β, ( z ( β, ω ) β ω ) ω ) ∈ e F we have ( α, ( z ( α, ω ) α ω ) ω ) ◦ ( β, ( z ( β, ω ) β ω ) ω ) = ( αβ, ( z ( α, βω ) α βω z ( β, ω ) β ω ) ω )= ( αβ, ( z ( α, βω ) z ( β, ω ) ◦ z ( β, ω ) − α βω z ( β, ω ) ◦ β ω ) ω )= ( αβ, ( z ( αβ, ω ) α ′ ω β ω ) ω ) ∈ e F for some ( α ′ ω ) ω ∈ K because Γ k ( F ) normalizes K . In particular, e F = h Γ k ( F ) , K i .We check condition (C) on these generators. As before, γ z ( z ( α, ω )) ∈ C ( γ z ( α ) , ω ) for all α ∈ F and ω ∈ Ω because z is involutive. Now, let k ∈ K . We then have γ z (pr ω k ) k ω ∈ C ( k, ω ) for all ω ∈ Ω by the assumption on k ω . (cid:3) In the split situation of Proposition 3.27 we also denote e F by Σ k ( F, K ) . Forinstance, the group Π( S , sgn , { } ) of Proposition 3.25 satisfies (C), admits an in-volutive compatibility cocycle but does not satisfy (D), see Section 4.3.Now, let F ≤ Sym(Ω) and ρ : F ։ A a homomorphism to an abelian group A .Further, let k ∈ N and X ⊆ { , . . . , k − } . Define Π k ( F, ρ, X ) := (cid:26) α ∈ Φ k ( F ) (cid:12)(cid:12)(cid:12)(cid:12) Y r ∈ X Y x ∈ S ( b,r ) ρ ( σ ( α, x )) = 1 (cid:27) . Proposition 3.28.
Let F ≤ Sym(Ω) and ρ : F ։ A a homomorphism to an abeliangroup A . Further, let k ∈ N and X ⊆ { , . . . , k − } non-empty and non-zero with k − ∈ X . If ρ ( F ω ) = A for all ω ∈ Ω then π (Π k ( F, ρ, X )) = F and Π k ( F, ρ, X ) has (C). Proof. As C F ( a, ω ) = aF ω , and ρ ( F ω ) = A for all ω ∈ Ω , an element α ∈ Φ k ( F ) canbe turned into an element of Π k ( F, ρ, X ) by changing σ ( α, x ) for a single, arbitrary x ∈ S ( b, k − . When X is non-zero we conclude that π (Π k ( F, ρ, X )) = F and that Π k ( F, ρ, X ) satisfies (C). (cid:3) A rigid case.
For certain F ≤ Sym(Ω) the groups Γ( F ) , ∆( F ) and Φ( F ) already yield all possible U k ( e F ) with π e F = F . The main argument is based onSections 3.4 and 3.5 of [BM00]. We first record the following lemma whose proof isdue to M. Giudici by personal communication. Lemma 3.29.
Let F ≤ Sym(Ω) be -transitive and F ω ( ω ∈ Ω) simple non-abelian.Then every extension e F of F ω ( ω ∈ Ω ) by F is equivalent to F ω × F . Proof.
Regarding F ω as a normal subgroup of e F , consider the conjugation map ϕ : e F → Aut( F ω ) . We show that K := ker ϕ = Z e F ( F ω ) E e F complements F ω in e F .Since Z ( F ω ) = { id } , we have F ω ∩ K = { id } . Hence F ω K E e F . Next, consider e F / ( F ω K ) . Out( F ω ) . By the solution of Schreier’ conjecture, Out( F ω ) is solvable.Since e F /F ω ∼ = F is not solvable we conclude K = { id } . Now, by a theorem ofBurnside, every -transitive permutation group F is either almost simple or affinetype, see [DM96, Theorem 4.1B and Section 4.8].In the first case, F is actually simple: Let N E F . Then F ω ∩ N E F ω . Henceeither F ω ∩ N = { id } or F ω ∩ N = F ω . Since F is -transitive and thereby primitive,every normal subgroup acts transitively. Hence, in the first case, N is regular whichcontradicts F being almost simple. Thus the second case holds and N = N F ω = F .Now e F /F ω K is a proper quotient of F and therefore trivial. We conclude that e F = F ω K ∼ = F ω × K and K ∼ = e F /F ω ∼ = F .In the second case, F = F ω ⋉ C dp for some d ∈ N and prime p . Given that K isnon-trivial and K ∼ = F ω K/F ω ⊳ ∼ F , it contains the unique minimal normal subgroup C dp ⊳ ∼ K ⊳ ∼ F . Since F/C dp ∼ = F ω is non-abelian simple whereas the proper quotient e F /F ω K of F is solvable, K = C dp . But F/C dp ∼ = F ω is simple, so F ω K = e F . (cid:3) The following propositions are of independent interest and used in Theorem 3.32below. We introduce the following notation: Let e F ≤ Aut( B d,k ) and K ≤ e F b w forsome w = ( ω , . . . , ω k − ) ∈ Ω ( k − . We set π w K := σ ( K, b w ) ≤ Sym(Ω) ω k − . Proposition 3.30.
Let e F ≤ Aut( B d,k ) satisfy (C). Suppose F := π e F is transitive.Further, let ω ∈ Ω and w = ( ω , . . . , ω k − ) ∈ Ω ( k − with ω = ω . Then π w ( e F b w ∩ ker π ) and π w e F T ω are subnormal in F ω k − of depth at most k − and k respectively. ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 23
Proof.
We argue by induction on k ≥ . For k = 2 , the assertion that π w ( e F b w ∩ ker π ) is normal in F ω is a consequence of condition (C). Now, suppose e F ≤ Aut( B d,k +1 ) satisfies the assumptions, and let ω ∈ Ω and w = ( ω , . . . , ω k ) ∈ Ω ( k ) be such that ω = ω . Since e F satisfies (C), we have pr ω ( e F b w ∩ ker π ) E ( π k e F ) b w ′ ∩ ker π , where w ′ := ( ω , . . . , ω k − ) and the right hand side π implicitly has domain π k e F . Hence π w ( e F b w ∩ ker π ) = π w ′ (pr ω ( e F b w ∩ ker π )) E π w ′ (( π k e F ) b w ′ ∩ ker π ) E F ω k − by the induction hypothesis. The second assertion follows as e F T ω E e F b w ∩ ker π . (cid:3) Proposition 3.31.
Let e F ≤ Aut( B d,k ) satisfy (C) but not (D). Suppose F := π e F is transitive, and every non-trivial subnormal subgroup of F ω ( ω ∈ Ω ) of depth atmost k − is transitive on Ω \{ ω } . Then U k ( e F ) is locally k -transitive. Proof.
We argue by induction on k . For k = 1 , the assertion follows from transitivityof F . Now, let e F ≤ Aut( B d,k +1 ) satisfy (C) but not (D). Then the same holds for F ( k ) := π k e F ≤ Aut( B d,k ) . Given e w, e w ′ ∈ Ω ( k ) , write e w = ( w, ω ) and e w ′ = ( w ′ , ω ′ ) where w, w ′ ∈ Ω ( k − and ω, ω ′ ∈ Ω . By the induction hypothesis, the group F ( k ) acts transitively on S ( b, k ) . Hence, using (C), there is g ∈ e F such that gb w = b w ′ .As e F does not satisfy (D) said transitivity further implies that π w ′ ( e F b w ′ ∩ ker π )) is non-trivial. By Proposition 3.30, it is also subnormal of depth at most k − in F ω ′ and thus transitive. Hence there is g ′ ∈ e F b w ′ with g ′ gb e w = b e w ′ . (cid:3) The following theorem is closely related to [BM00, Proposition 3.3.1].
Theorem 3.32.
Let F ≤ Sym(Ω) be -transitive and F ω ( ω ∈ Ω) simple non-abelian.Further, let e F ≤ Aut( B d,k ) with π e F = F satisfy (C). Then U k ( e F ) equals either U (Γ( F )) , U (∆( F )) or U (Φ( F )) = U ( F ) . Proof.
Since U ( F ) = U (Φ( F )) , we may assume k ≥ . Given that e F ≤ Aut( B d,k ) satisfies (C) so does the restriction F (2) := π e F ≤ Φ( F ) ≤ Aut( B d, ) . Consider theprojection π : F (2) ։ F . We have ker π ≤ Q ω ∈ Ω F ω and pr ω ker π E F ω for all ω ∈ Ω by Proposition 3.30. Since F ω is simple, ker π E F (2) and F is transitive this impliesthat either pr ω ker π = { id } for all ω ∈ Ω or pr ω ker π = F ω for all ω ∈ Ω .In the first case, π : F (2) → F is an isomorphism. Hence F (2) satisfies (CD) and U k ( e F ) = U (Γ( F )) for an involutive compatibility cocycle of F by Proposition 3.26.In the second case, fix ω ∈ Ω . We have ker π ≤ Q ω ∈ Ω F ω ∼ = F dω by transitivityof F . Since F ω is simple non-abelian, [Rad17, Lemma 2.3] implies that the group ker π is a product of subdiagonals preserved by the primitive action of F on theindex set of F dω . Hence, either there is just one block and ker π ∼ = F ω has the form { (id , ( α ω ( a )) ω ) | a ∈ F ω } for some isomorphisms α ω : F ω → F ω , or all blocksare singletons and ker π = Q ω ∈ Ω F ω ∼ = F dω . In the first case, there is a compatibilitycocycle z of F such that F ∼ = { ( a, ( z ( a, ω )) ω ) | a ∈ F } ≤ F (2) commutes with ker π ≤ F (2) by Lemma 3.29. Thus F (2) = { a, ( z ( a, ω ) α ω ( a )) ω | a ∈ F, a ∈ F ω } .In particular, F (2) satisfies (CD). Hence U k ( e F ) = U (∆( F )) .When ker π ∼ = F dω , we have U k ( e F ) = U ( F ) by [BM00, Proposition 3.3.1]. (cid:3) If F does not have simple point stabilizers or preserves a non-trivial partition,more universal groups are given by U (Φ( F, N )) and U (Φ( F, P )) , see Section 3.4.1.When F is -transitive and has abelian point stabilizers then F ∼ = AGL(1 , q ) for someprime power q by [KKP90]. Hence point stabilizers in F are isomorphic to F ∗ q andsimple if and only if q − is a Mersenne prime. For any value of q , the projection ρ : AGL(1 , q ) → F ∗ q satisfies the assumptions of Proposition 3.28 and so the groups U k (Π k (AGL(1 , q ) , ρ, X )) provide further examples. The following question remains. Question 3.33.
Let F ≤ Sym(Ω) be primitive and F ω ( ω ∈ Ω) simple non-abelian.Is there e F ≤ Aut( B d,k ) satisfying (C) and π e F = F other than Γ( F ) , ∆( F ) and Φ( F ) ? Universality.
The constructed groups U k ( F ) are universal in the sense of thefollowing maximality statement, which should be compared to Proposition 1.6. Theorem 3.34.
Let H ≤ Aut( T d ) be locally transitive and contain an involutiveinversion. Then there is a labelling l of T d such that U ( l )1 ( F (1) ) ≥ U ( l )2 ( F (2) ) ≥ · · · ≥ U ( l ) k ( F ( k ) ) ≥ · · · ≥ H ≥ U ( l )1 ( { id } ) where F ( k ) ≤ Aut( B d,k ) is action isomorphic to the k -local action of H . Proof.
First, we construct a labelling l of T d such that H ≥ U ( l )1 ( { id } ) : Fix x ∈ V and choose a bijection l x : E ( x ) → Ω . By the assumptions, there is an involutiveinversion ι ω ∈ H of the edge ( x, x ω ) ∈ E for every ω ∈ Ω . Using these inversions,we define the announced labelling inductively: Set l | E ( x ) := l x and assume that l isdefined on E ( x, n ) . For e ∈ E ( x, n + 1) \ E ( x, n ) put l ( e ) := l ( ι ω ( e )) if x ω is part ofthe unique reduced path from x to o ( e ) . Since the ι ω ( ω ∈ Ω) have order , we obtain σ ( ι ω , y ) = id for all ω ∈ Ω and y ∈ V . Therefore, h{ ι ω | ω ∈ Ω }i = U ( l )1 ( { id } ) ≤ H ,following the proof of Lemma 1.5.Now, let h ∈ H and y ∈ V . Further, let ( x, x , . . . , x n , y ) and ( x, x ′ , . . . , x ′ m , h ( y )) be the unique reduced paths from x to y and h ( y ) respectively. Since U ( l )1 ( { id } ) ≤ H ,the group H contains the unique label-respecting inversion ι e of every edge e ∈ E .We therefore have s := ι − x ′ ,x ) · · · ι − x ′ m ,x ′ m − ) ι − h ( y ) ,x ′ m ) ◦ h ◦ ι ( y,x n ) · · · ι ( x ,x ) ι ( x ,x ) ∈ H, Also, s stabilizes x . The cocycle identity implies for every k ∈ N : σ k ( h, y ) = σ k ( ι ( h ( y ) ,x ′ m ) · · · ι ( x ′ ,x ) ◦ s ◦ ι − x ,x ) · · · ι − y,x n ) , y ) = σ k ( s, x ) ∈ F ( k ) . where F ( k ) ≤ Aut( B d,k ) is defined by l kx ◦ H x | B ( x,k ) ◦ ( l kx ) − . (cid:3) Remark 3.35.
Retain the notation of Theorem 3.34. By Proposition 1.6, thereis a labelling l of T d such that U ( l )1 ( F (1) ) ≥ H regardless of the minimal order ofan inversion in H . This labelling may be distinct from the one of Theorem 3.34which fails without assuming the existence of an involutive inversion: For example,a vertex-stabilizer of the group G of Example 4.39 below is action isomorphic to Γ( S ) but G U ( l )2 (Γ( S )) for any labelling l because ( G ) { b,b ω } ∼ = Z / Z whereas U ( l )2 (Γ( S )) { b,b ω } ∼ = Γ( S ) ( b,b ω ) ⋊ Z / Z ∼ = Z / Z × Z / Z by Proposition 3.14.We complement Theorem 3.34 with the following criterion for certain subgroupsof Aut( T d ) to contain an involutive inversions. Proposition 3.36.
Let H ≤ Aut( T d ) be locally transitive with odd order pointstabilizers. If H contains a finite order inversion then it contains an involutive one. Proof.
Let ι ∈ H be a finite order inversion of an edge e ∈ E and ord( ι ) = 2 k · m forsome odd m ∈ N and some k ∈ N . It suffices to show that k = 1 , in which case ι m is an involutive inversion. Suppose k ≥ . Then ι k − · m is non-trivial and fixes theedge e . Because point stabilizers in the local action of H have odd order, it followsthat ( ι k − · m ) is non-trivial as well, but ( ι k − · m ) = ι ord( ι ) = id . (cid:3) For example, Proposition 3.36 applies when H is discrete and vertex-transitive:Combined with local transitivity this implies the existence of a finite order inversion.We remark that primitive permutation groups with odd order point stabilizerswere classified in [LS91]. For instance, they include PSL(2 , q ) y P ( F q ) for anyprime power q that satisfies q ≡ . ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 25
A Bipartite Version.
In this section, we introduce a bipartite version ofthe universal groups developed in Section 3 which plays a critical role in the proofof Theorem 4.2(iv)(b) below. Retain the notation of Section 3. In particular, let T d = ( V, E ) denote the d -regular tree. Fix a regular bipartition V = V ⊔ V of V .3.6.1. Definition and Basic Properties.
The groups to be defined are subgroups of
Aut( T d ) + ≤ Aut( T d ) , the maximal subgroup of Aut( T d ) preserving the bipartition V = V ⊔ V . Alternatively, it can be described as the subgroup generated by allpoint stabilizers, or all edge-stabilizers. Definition 3.37.
Let F ≤ Aut( B d, k ) and l be a labelling of T d . Define BU ( l )2 k ( F ) := { α ∈ Aut( T d ) + | ∀ v ∈ V : σ k ( α, v ) ∈ F } . Note that BU ( l )2 k ( F ) is a subgroup of Aut( T d ) + thanks to Lemma 3.2 and theassumption that it is a subset of Aut( T d ) + . Further, Proposition 3.4 carries over tothe groups BU ( l )2 k ( F ) . We shall therefore omit the reference to an explicit labellingin the following. Also, we recover the following basic properties. Proposition 3.38.
Let F ≤ Aut( B d, k ) . The group BU k ( F ) is(i) closed in Aut( T d ) (ii) transitive on both V and V , and(iii) compactly generated.Parts (i) and (ii) are proven as their analogues in Proposition 3.5 whereas part(iii) relies on part (ii) and the subsequent analogue of Lemma 1.5, for which weintroduce the following notation: Given x ∈ V and w ∈ Ω (2 k ) , let t ( x ) w ∈ BU ( { id } ) denote the unique label-respecting translation with t ( x ) w ( x ) = x w . Given an element w = ( ω , . . . , ω k ) ∈ Ω (2 k ) , we set w := ( ω k , . . . , ω ) ∈ Ω (2 k ) . Then ( t ( x ) w ) − = t ( x ) w and if Ω (2 k )+ ⊆ Ω (2 k ) is such that for every w ∈ Ω (2 k ) exactly one of { w, w } belongsto Ω (2 k )+ , then Ω (2 k )+ = Ω (2 k )+ ⊔ Ω (2 k )+ where Ω (2 k )+ := { w | w ∈ Ω (2 k )+ } . Lemma 3.39.
Let x ∈ V . Then BU ( { id } ) = h{ t ( x ) w | w ∈ Ω (2) }i ∼ = F Ω (2)+ , the freegroup on the set Ω (2)+ . Proof.
Every element of BU k ( { id } ) is uniquely determined by its image on x . To seethat BU ( { id } ) = h{ t ( x ) w | w ∈ Ω (2) }i it hence suffices to show that { t ( x ) w | w ∈ Ω (2) } is transitive on V . Indeed, let y ∈ V . Then y = x w for some w ∈ Ω (2 k ) where k = d ( x, y ) . Write w = ( w , . . . , w k ) ∈ (Ω (2) ) k . Then t ( x ) w ◦ · · · ◦ t ( x ) w k = t ( x ) w as every t ( x ) w i ( i ∈ { , . . . , k } ) is label-respecting. Hence t ( x ) w ◦ · · · ◦ t ( x ) w k ( x ) = y and that h{ t ( x ) w | w ∈ Ω (2) }i → F Ω (2)+ , ( t ( x ) w w w ∈ Ω (2)+ t ( x ) w w − w Ω (2)+ yields a well-defined isomorphism. (cid:3) Compatibility and Discreteness.
In order to describe the compatibility andthe discreteness condition in the bipartite setting, we first introduce a suitablerealization of
Aut( B d, k ) ( k ∈ N ) , similar to the one at the beginning of Section 3.4.Let Aut( B d, ) ∼ = Sym(Ω) and Aut( B d, ) be as before. For k ≥ , we inductivelyidentify Aut( B d, k ) with its image under Aut( B d, k ) → Aut( B d, k − ) ⋉ Y w ∈ Ω (2) Aut( B d, k − ) α ( σ k − ( α, b ) , ( σ k − ( α, b w )) w )) where Aut( B d, k − ) acts on Ω (2) by permuting the factors according to its actionon S ( b, ∼ = Ω (2) . In addition, consider the map pr w : Aut( B d, k ) → Aut( B d, k − ) , α σ k − ( α, b w ) for every w ∈ Ω (2) , as well as p w = ( π k − , pr w ) : Aut( B d, k ) → Aut( B d, k − ) × Aut( B d, k − ) For k ≥ , conditions (C) and (D) for F ≤ Aut( B d, k ) now read as follows.(C) ∀ α ∈ F ∀ w ∈ Ω (2) ∃ α w ∈ F : π k − ( α w ) = pr w ( α ) , pr w ( α w ) = π k − ( α ) (D) ∀ w ∈ Ω (2) : p w | − F (id , id) = { id } For k = 1 we have, using the maps pr ω ( ω ∈ Ω ) as in Section 3.4,(C) ∀ α ∈ F ∀ w = ( ω , ω ) ∈ Ω (2) ∃ α w ∈ F : pr ω ( α w ) = pr ω ( α ) . (D) ∀ ω ∈ Ω : pr ω | − F (id) = { id } . Analogues of Proposition 3.12 are proven using the discreteness conditions (D)above. We do not introduce new notation for any of the above as the context alwaysimplies which condition is to be considered. The definition of the compatibilitysets C F ( α, S ) for F ≤ Aut( B d, k ) and S ⊆ Ω (2) carries over from Section 3.2 in astraightforward fashion.3.6.3. Examples.
Let F ≤ Sym(Ω) . Then the group Γ( F ) ≤ Aut( B d, ) introducedin Section 3.4.1 satisfies conditions (C) and (D) for the case k = 1 above, and wehave BU (Γ( F )) = U (Γ( F )) ∩ Aut( T d ) + .Similarly, the group Φ( F ) ≤ Aut( B d, ) satisfies condition (C) for the case k = 1 as Γ( F ) ≤ Φ( F ) , and we have BU (Φ( F )) = U ( F ) ∩ Aut( T d ) + .The following example gives an analogue of the groups Φ( F, N ) . Notice, however,that in this case the second argument is a subgroup of F rather than F ω and neednot be normal, as the -local action at vertices in V and V need not be the same. Example 3.40.
Let F ′ ≤ F ≤ Sym(Ω) . Then
BΦ(
F, F ′ ) := { ( a, ( a ω ) ω ∈ Ω ) | a ∈ F, ∀ ω ∈ Ω : a ω ∈ C F ( a, ω ) ∩ F ′ } ≤ Aut( B d, ) satisfies condition (C) for the case k = 1 above given that Γ( F ′ ) ≤ BΦ(
F, F ′ ) . If F ′ \ Ω = F \ Ω , the -local action of BΦ(
F, F ′ ) at vertices in V is indeed F , whereasit is F ′ + at vertices in V . This construction is similar to U L ( M, N ) in [Smi17].The next example constitutes the base case in Section 4.1.5 below. Example 3.41.
Let F ≤ Sym(Ω) . Suppose F preserves a non-trivial partition P : Ω = F i ∈ I Ω i of Ω . Then Ω (2)0 := { ( ω , ω ) | ∃ i ∈ I : ω , ω ∈ Ω i } ⊆ Ω (2) . is preserved by the action of Φ( F ) on S ( b, ∼ = Ω (2) : Let α = ( a, ( a ω ) ω ) ∈ Φ( F ) and ( ω , ω ) ∈ Ω (2)0 . Then α ( ω , ω ) = ( aω , a ω ω ) = ( a ω ω , a ω ω ) ∈ Ω (2)0 . Also, notethat if w = ( ω , ω ) ∈ Ω (2)0 then so is w = ( ω , ω ) .The subgroup of Φ( F ) consisting of those elements which are self-compatible inall directions from Ω (2)0 is precisely given by F (2) := { ( a, ( a ω ) ω ) | a ∈ F, a ω ∈ C F ( a, ω ) constant w.r.t. P} . in view of condition (C) for the case k = 1 above.Suppose now that F ≤ Aut( B d, k ) satisfies (C). Analogous to the group Φ k ( F ) of Section 3.4.2, we define BΦ k ( F ) := { ( α, ( α w ) w ∈ Ω (2) ) | α ∈ F, ∀ w ∈ Ω (2) : α w ∈ C F ( α, w ) } ≤ Aut( B d, k +1) ) . Then BΦ k ( F ) ≤ Aut( B d, k +1) ) satisfies (C) and BU k +1) (BΦ k ( F )) = BU k ( F ) .Given l > k , we also set BΦ l ( F ) := BΦ l − ◦ · · · ◦ BΦ k ( F ) , c.f. Section 3.4.2. ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 27 Applications
In this section, we give three applications of the framework of universal groups.First, we characterize the automorphism types which the quasi-center of a non-discrete subgroup of
Aut( T d ) may feature in terms of the group’s local action, andsee that Burger–Mozes theory does not extend to the transitive case. Second, wegive an algebraic characterization of the k -closures of locally transitive subgroupsof Aut( T d ) containing an involutive inversion, and thereby partially answer twoquestion by Banks–Elder–Willis. Third, we offer a new view on the Weiss conjecture.4.1. Groups Acting on Trees With Non-Trivial Quasi-Center.
By Propo-sition 2.11(ii), a non-discrete, locally semiprimitive subgroup of
Aut( T d ) does notcontain any non-trivial quasi-central edge-fixating elements. We complete this factto the following local-to-global-type characterization of quasi-central elements. Theorem 4.1.
Let H ≤ Aut( T d ) be non-discrete. If H is locally(i) transitive then QZ( H ) contains no inversion.(ii) semiprimitive then QZ( H ) contains no non-trivial edge-fixating element.(iii) quasiprimitive then QZ( H ) contains no non-trivial elliptic element.(iv) k -transitive ( k ∈ N ) then QZ( H ) contains no hyperbolic element of length k . Theorem 4.2.
There is d ∈ N ≥ and a closed, non-discrete, compactly generatedsubgroup of Aut( T d ) which is locally(i) intransitive and contains a quasi-central inversion.(ii) transitive and contains a non-trivial quasi-central edge-fixating element.(iii) semiprimitive and contains a non-trivial quasi-central elliptic element.(iv) (a) intransitive and contains a quasi-central hyperbolic element of length .(b) quasiprimitive and contains a quasi-central hyperbolic element of length . Proof. (Theorem 4.1). Fix a labelling of T d and let H ≤ Aut( T d ) be non-discrete.For (i), suppose ι ∈ QZ( H ) inverts ( x, x ω ) ∈ E . Since H is locally transitiveand QZ( H ) E H , there is an inversion ι ω ∈ QZ( H ) of ( x, x ω ) ∈ E for all ω ∈ Ω .By definition, the centralizer of ι ω in H is open for all ω ∈ Ω . Hence, using non-discreteness of H , there is n ∈ N such that H B ( x,n ) commutes with ι ω for all ω ∈ Ω and H B ( x,n +1) = { id } . However, H B ( x,n ) = ι ω H B ( x,n ) ι − ω = H B ( x ω ,n ) for all ω ∈ Ω ,that is H B ( x,n +1) ⊆ H B ( x,n ) in contradiction to the above.Part (ii) is Proposition 2.11(ii) and part (iii) is [BM00, Proposition 1.2.1 (ii)].Here, the closedness assumption is unnecessary.For part (iv), suppose τ ∈ QZ( H ) is a translation of length k which maps x ∈ V to x w ∈ V for some w ∈ Ω ( k ) . Since H is locally k -transitive and QZ( H ) E H , thereis a translation τ w ∈ QZ( H ) which maps x to x w for all w ∈ Ω ( k ) . By definition,the centralizer of τ w in H is open for all w ∈ Ω ( k ) . Hence, using non-discretenessof H there is n ∈ N such that H B ( x,n ) commutes with τ w for all w ∈ Ω ( k ) and H B ( x,n +1) = { id } . However, H B ( x,n ) = τ w H B ( x,n ) τ − w = H B ( x w ,n ) for all w ∈ Ω ( k ) ,that is H B ( x,n + k ) ⊆ H B ( x,n ) in contradiction to the above. (cid:3) We complement part (ii) of Theorem 4.1 with the following result inspired by[BM00, Proposition 3.1.2] and [Rat04, Conjecture 2.63],
Proposition 4.3.
Let H ≤ Aut( T d ) be non-discrete and locally semiprimitive. Ifall orbits of H y ∂T d are uncountable then QZ( H ) is trivial. Proof.
By Theorem 4.1, the group
QZ( H ) contains no inversions. Let S ⊆ ∂T d bethe set of fixed points of hyperbolic elements in QZ( H ) . Since QZ( H ) E H , the set S is H -invariant. Also, QZ( H ) is discrete by Theorem 4.1 and hence countable as H is second-countable. Thus S is countable and hence empty. We conclude that QZ( H ) E H does not contain elliptic elements in view of [GGT18, Lemma 6.4]. (cid:3) The following strengthening of Theorem 4.2(ii) proved in Section 4.1.2 showsthat Burger–Mozes theory does not generalize to the locally transitive case.
Theorem 4.4.
There is d ∈ N ≥ and a closed, non-discrete, compactly generated,locally transitive subgroup of Aut( T d ) with open, hence non-discrete, quasi-center.We prove Theorem 4.2 by construction in the consecutive sections. Whereasparts (i) to (iv)(a) all use groups of the form T k ∈ N U k ( F ( k ) ) for appropriate localactions F ( k ) ≤ Aut( B d,k ) , part (iv)(b) uses a group of the form T k ∈ N BU( F (2 k ) ) .All sections appear similar at first glance but vary in detail.4.1.1. Theorem 4.2(i).
For certain intransitive F ≤ Sym(Ω) we construct a closed,non-discrete, compactly generated, vertex-transitive group H ( F ) ≤ Aut( T d ) whichlocally acts like F and contains a quasi-central involutive inversion.Let F ≤ Sym(Ω) . Assume that the partition F \ Ω = F i ∈ I Ω i of Ω into F -orbitshas at least three elements and that F Ω i = { id } for all i ∈ I .Fix an orbit Ω of size at least and ω ∈ Ω . Define groups F ( k ) ≤ Aut( B d,k ) for k ∈ N inductively by F (1) := F and F ( k +1) := { ( α, ( α ω ) ω ) | α ∈ F ( k ) , α ω ∈ C F ( k ) ( α, ω ) constant w.r.t. F \ Ω , α ω = α } . Proposition 4.5.
The groups F ( k ) ≤ Aut( B d,k ) ( k ∈ N ) defined above satisfy:(i) Every α ∈ F ( k ) is self-compatible in directions from Ω .(ii) The compatibility set C F ( k ) ( α, Ω i ) is non-empty for all α ∈ F ( k ) and i ∈ I .In particular, the group F ( k ) satisfies (C).(iii) The compatibility set C F ( k ) (id , Ω i ) is non-trivial for all Ω i = Ω .In particular, the group F ( k ) does not satisfy (D). Proof.
We prove all three properties simultaneously by induction: For k = 1 , theassertions (i) and (ii) are trivial. The third translates to F Ω i being non-trivial forall Ω i = Ω which is an assumption. Now, assume that all properties hold for F ( k ) .Then the definition of F ( k +1) is meaningful because of (i) and it is a subgroup of Aut( B d,k +1 ) because F preserves each Ω i ( i ∈ I ) . Assertion (i) is now evident.Statement (ii) carries over from F ( k ) to F ( k +1) . So does (iii) since | F \ Ω | ≥ . (cid:3) Definition 4.6.
Retain the above notation. Define H ( F ) := T k ∈ N U k ( F ( k ) ) .Now, H ( F ) is compactly generated, vertex-transitive and contains an involutiveinversion because U ( { id } ) ≤ H ( F ) . Also, H ( F ) is closed as an intersection of closedsets. The -local action of H is given by F = F (1) because Γ k ( F ) ≤ F ( k ) for all k ∈ N and therefore D( F ) ≤ H ( F ) . Lemma 4.7.
The group H ( F ) is non-discrete. Proof.
Let x ∈ V and n ∈ N . We construct a non-trivial element h ∈ H ( F ) whichfixes B ( x, n ) : Set α n := id ∈ F ( n ) . By parts (i) and (iii) of Proposition 4.5 aswell as the definition of F ( n +1) , there is a non-trivial element α n +1 ∈ F ( n +1) with π n α n +1 = α n . Applying parts (i) and (ii) of Proposition 4.5 repeatedly, we obtainnon-trivial elements α k ∈ F ( k ) for all k ≥ n +1 with π k α k +1 = α k . Set α k := id ∈ F ( k ) for all k ≤ n and define h ∈ Aut( T d ) x by fixing x and setting σ k ( h, x ) := α k ∈ F ( k ) .Since F ( l ) ≤ Φ l ( F ( k ) ) for all k ≤ l we conclude that h ∈ T k ∈ N U k ( F ( k ) ) = H ( F ) . (cid:3) Proposition 4.8.
The quasi-center of H ( F ) contains an involutive inversion. Proof.
Let x ∈ V . The group QZ( H ( F )) contains the label-respecting inversion ι ω of ( x, x ω ) ∈ E for all ω ∈ Ω : Let h ∈ H ( F ) B ( x, and ω ∈ Ω . Then hι ω ( x ) = x ω = ι ω h ( x ) and σ k ( hι ω , x ) = σ k ( h, ι ω x ) σ k ( ι ω , x ) = σ k ( h, x ω ) = σ k ( ι ω , hx ) σ k ( h, x ) = σ k ( ι ω h, x ) for all k ∈ N since h ∈ U k +1 ( F ( k +1) ) . That is, ι ω commutes with H ( F ) B ( b, . (cid:3) ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 29
Theorem 4.2(ii).
For certain transitive F ≤ Sym(Ω) we construct a closed,non-discrete, compactly generated, vertex-transitive group H ( F ) ≤ Aut( T d ) whichlocally acts like F and has open quasi-center.Let F ≤ Sym(Ω) be transitive. Assume that F preserves a non-trivial partition P : Ω = F i ∈ I Ω i of Ω and that F Ω i = { id } for all i ∈ I . Further, suppose that F + is abelian and preserves P setwise. Example 4.9.
Let F ′ ≤ Sym(Ω ′ ) be regular abelian and P ≤ Sym(Λ) regular.Then F := F ′ ≀ P ≤ Sym(Ω ′ × Λ) satisfies the above properties as F + = Q λ ∈ Λ F ′ .Define groups F ( k ) ≤ Aut( B d,k ) for k ∈ N inductively by F (1) := F and F ( k +1) := { ( α, ( α ω ) ω ) | α ∈ F ( k ) , α ω ∈ C F ( k ) ( α, ω ) constant w.r.t. P} . Proposition 4.10.
The groups F ( k ) ≤ Aut( B d,k ) ( k ∈ N ) defined above satisfy:(i) The compatibility set C F ( k ) ( α, Ω i ) is non-empty for all α ∈ F ( k ) and i ∈ I .In particular, the group F ( k ) satisfies (C).(ii) The compatibility set C F ( k ) (id , Ω i ) is non-trivial for all i ∈ I .In particular, the group F ( k ) does not satisfy (D).(iii) The group F ( k ) ∩ Φ k ( F + ) is abelian. Proof.
We prove all three properties simultaneously by induction: For k = 1 , theassertion (i) is trivial whereas (iii) is an assumption. The second translates to F Ω i being non-trivial for all i ∈ I which is an assumption. Now, assume all propertieshold for F ( k ) . Then the definition of F ( k +1) is meaningful because of (i) and it is asubgroup of Aut( B d,k ) because F preserves P . Statement (ii) carries over from F ( k ) to F ( k +1) . Finally, (iii) follows inductively because F + preserves P setwise. (cid:3) Definition 4.11.
Retain the above notation. Define H ( F ) := T k ∈ N U k ( F ( k ) ) .Now, H ( F ) is compactly generated, vertex-transitive and contains an involutiveinversion because U ( { id } ) ≤ H ( F ) . Also, H ( F ) is closed as an intersection of closedsets. The -local action of H is given by F = F (1) because Γ k ( F ) ≤ F ( k ) for all k ∈ N and therefore D( F ) ≤ H ( F ) . Lemma 4.12.
The group H ( F ) is non-discrete. Proof.
Let x ∈ V and n ∈ N . We construct a non-trivial element h ∈ H ( F ) which fixes B ( x, n ) : Consider α n := id ∈ F ( n ) . By part (ii) of Proposition 4.10 aswell as the definition of F ( n +1) , there is a non-trivial element α n +1 ∈ F ( n +1) with π n α n +1 = α n . Applying part (i) of Proposition 4.10 repeatedly, we obtain non-trivial elements α k ∈ F ( k ) for all k ≥ n + 1 with π k α k +1 = α k . Set α k := id ∈ F ( k ) for all k ≤ n and define h ∈ Aut( T d ) x by fixing x and setting σ k ( h, x ) := α k ∈ F ( k ) .Since F ( l ) ≤ Φ l ( F ( k ) ) for all k ≤ l we conclude that h ∈ T k ∈ N U k ( F ( k ) ) = H ( F ) . (cid:3) Proposition 4.13.
The group H ( F ) has open quasi-center. Proof.
The group H ( F ) B ( x, is a subgroup of the group H ( F + ) x which is abelianby part (iii) of Proposition 4.10. Hence H ( F ) B ( x, ≤ QZ( H ( F )) . (cid:3) Remark 4.14.
Without assuming local transitivity one can achieve abelian pointstabilizers, following the construction of the previous section. This cannot happenfor non-discrete locally transitive groups H ≤ Aut( T d ) which are vertex-transitiveas the following argument shows: By Proposition 1.6, the group H is contained in U( F ) where F ≤ Sym(Ω) is the local action of H . If H x is abelian, then so is F .Since any transitive abelian permutation group is regular we conclude that U( F ) and hence H are discrete. In this sense, the construction of this section is efficient. Theorem 4.2(iii).
For certain semiprimitive F ≤ Sym(Ω) we construct aclosed, non-discrete, compactly generated, vertex-transitive group H ( F ) ≤ Aut( T d ) which locally acts like F and contains a non-trivial quasi-central elliptic element.Let F ≤ Sym(Ω) be semiprimitive. Suppose F preserves a non-trivial partition P : Ω = F i ∈ I Ω i of Ω and that F Ω i = { id } for all i ∈ I . Further, suppose that F contains a non-trivial central element τ which preserves P setwise. Example 4.15.
Consider
SL(2 , y F \{ } = {± e , ± e , ± ( e + e ) , ± ( e − e ) } where e , e are the standard basis vectors. We have Z (SL(2 , {± Id } . Theblocks of size are as listed above given that SL(2 , e ≤ ± SL(2 , e .Define groups F ( k ) ≤ Aut( B d,k ) for k ∈ N inductively by F (1) := F and F ( k +1) := { ( α, ( α ω ) ω ) | α ∈ F ( k ) , α ω ∈ C F ( k ) ( α, ω ) constant w.r.t P} . Proposition 4.16.
The groups F ( k ) ≤ Aut( B d,k ) ( k ∈ N ) defined above satisfy:(i) The compatibility set C F ( k ) ( α, Ω i ) is non-empty for all α ∈ F ( k ) and i ∈ I .In particular, the group F ( k ) satisfies (C).(ii) The compatibility set C F ( k ) (id , Ω i ) is non-trivial for all i ∈ I .In particular, the group F ( k ) does not satisfy (D).(iii) The element γ k ( τ ) ∈ Aut( B d,k ) is central in F ( k ) . Proof.
We prove all three properties simultaneously by induction: For k = 1 , theassertion (i) is trivial whereas (iii) is an assumption. The second translates to F Ω i being non-trivial for all i ∈ I which is an assumption. Now, assume all propertieshold for F ( k ) . Then the definition of F ( k +1) is meaningful because of (i) and it is asubgroup of Aut( B d,k +1 ) because F preserves P . Statement (ii) carries over from F ( k ) to F ( k +1) . Finally, (iii) follows inductively because τ and hence τ − preserves P setwise: For e α = ( α, ( α ω ) ω ) ∈ F ( k +1) we have γ k +1 ( τ ) e αγ k +1 ( τ ) − = ( γ k ( τ ) αγ k ( τ ) − , ( γ k ( τ ) α τ − ( ω ) γ k ( τ ) − ) ω ) . (cid:3) Definition 4.17.
Retain the above notation. Define H ( F ) := T k ∈ N U k ( F ( k ) ) .Now, H ( F ) is compactly generated, vertex-transitive and contains an involutiveinversion because U ( { id } ) ≤ H ( F ) . Also, H ( F ) is closed as an intersection of closedsets. The -local action of H is given by F = F (1) because Γ k ( F ) ≤ F ( k ) for all k ∈ N and therefore D( F ) ≤ H ( F ) . Lemma 4.18.
The group H ( F ) is non-discrete. Proof.
Let x ∈ V and n ∈ N . We construct a non-trivial element h ∈ H ( F ) whichfixes B ( x, n ) : Consider α n := id ∈ F ( n ) . By part (ii) of Proposition 4.16 and thedefinition of F ( n +1) , there is a non-trivial α n +1 ∈ F ( n +1) with π n α n +1 = α n .Applying part (i) of Proposition 4.16 repeatedly, we obtain non-trivial elements α k ∈ F ( k ) for all k ≥ n + 1 with π k α k +1 = α k . Set α k := id ∈ F ( k ) for all k ≤ n and define h ∈ Aut( T d ) x by fixing x and setting σ k ( h, x ) := α k ∈ F ( k ) . Since F ( l ) ≤ Φ l ( F ( k ) ) for all k ≤ l we conclude that h ∈ T k ∈ N U k ( F ( k ) ) = H ( F ) . (cid:3) Proposition 4.19.
The quasi-center of H ( F ) contains a non-trivial elliptic element. Proof.
By Proposition 4.16, the element d ( τ ) which fixes x and whose -local actionis τ everywhere commutes with H ( F ) x . Hence d ( τ ) ∈ QZ( H ( F )) . (cid:3) Remark 4.20.
The argument of this section does not work in the quasiprimitivecase because a quasiprimitive group F ≤ Sym(Ω) with non-trivial center is abelianand regular: If Z ( F ) E F is non-trivial then it is transitive, and it suffices to showthat F + is trivial. Suppose a ∈ F ω moves ω ′ ∈ Ω . Pick z ∈ Z ( F ) with z ( ω ) = ω ′ .Then za ( ω ) = ω ′ = az ( ω ) , contradicting the assumption that z ∈ Z ( F ) . ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 31
Theorem 4.2(iv)(a).
For certain intransitive F ≤ Sym(Ω) we construct aclosed, non-discrete, compactly generated, vertex-transitive group H ( F ) ≤ Aut( T d ) which locally acts like F and contains a quasi-central hyperbolic element of length .Let F ≤ Sym(Ω) . Assume that the partition F \ Ω = F i ∈ I Ω i of Ω has at leastthree elements and that Z ( F ) = { id } . Choose a non-trivial element τ ∈ Z ( F ) and ω ∈ Ω ∈ F \ Ω with τ ( ω ) = ω . Further, suppose that F Ω i = { id } for all Ω i = Ω .Define groups F ( k ) ≤ Aut( B d,k ) for k ∈ N inductively by F (1) := F and F ( k +1) := { ( α, ( α ω ) ω ) | α ∈ F ( k ) , α ω ∈ C F ( k ) ( α, ω ) constant w.r.t. F \ Ω , α ω = α } . Proposition 4.21.
The groups F ( k ) ≤ Aut( B d,k ) ( k ∈ N ) defined above satisfy:(i) Every α ∈ F ( k ) is self-compatible in directions from Ω .(ii) The compatibility set C F ( k ) ( α, Ω i ) is non-empty for all α ∈ F ( k ) and i ∈ I .In particular, the group F ( k ) satisfies (C).(iii) The compatibility set C F ( k ) (id , Ω i ) is non-trivial for all i ∈ I \{ } .In particular, the group F ( k ) does not satisfy (D).(iv) The element γ k ( τ ) ∈ Aut( B d,k ) is central in F ( k ) . Proof.
We prove all four properties simultaneously by induction: For k = 1 , theassertions (i) and (ii) are trivial. The third translates to F Ω i being non-trivial for all i ∈ I \{ } which is an assumption, as is (iv). Now, assume that all properties hold for F ( k ) . Then the definition of F ( k +1) is meaningful because of (i) and it is a subgroupof Aut( B d,k ) because F preserves F \ Ω . Assertion (i) is now evident. Statements (ii)and (iii) carry over from F ( k ) to F ( k +1) . Finally, (iii) follows inductively because τ and hence τ − preserves F \ Ω setwise: For e α = ( α, ( α ω ) ω ) ∈ F ( k +1) we have γ k +1 ( τ ) e αγ k +1 ( τ ) − = ( γ k ( τ ) αγ k ( τ ) − , ( γ k ( τ ) α τ − ( ω ) γ k ( τ ) − ) ω ) . (cid:3) Definition 4.22.
Retain the above notation. Define H ( F ) := T k ∈ N U k ( F ( k ) ) .Now, H ( F ) is compactly generated, vertex-transitive and contains an involutiveinversion because U ( { id } ) ≤ H ( F ) . Also, H ( F ) is closed as the intersection of allits k -closures. The -local action of H is given by F = F (1) as Γ k ( F ) ≤ F ( k ) for all k ∈ N and therefore D( F ) ≤ H . Lemma 4.23.
The group H ( F ) is non-discrete. Proof.
Let x ∈ V and n ∈ N . We construct a non-trivial element h ∈ H ( F ) whichfixes B ( x, n ) : Consider α n := id ∈ F ( n ) . By parts (i) and (iii) of Proposition 4.21as well as the definition of F ( n +1) , there is a non-trivial element α n +1 ∈ F ( n +1) with π n α n +1 = α n . Applying parts (i) and (ii) of Proposition 4.21 repeatedly, weobtain non-trivial elements α k ∈ F ( k ) for all k ≥ n + 1 with π k α k +1 = α k . Set α k := id ∈ F ( k ) for all k ≤ n and define h ∈ Aut( T d ) x by fixing x and setting σ k ( h, x ) := α k ∈ F ( k ) . Since F ( l ) ≤ Φ l ( F ( k ) ) for all k ≤ l we conclude that h ∈ T k ∈ N U k ( F ( k ) ) = H ( F ) . (cid:3) Proposition 4.24.
The quasi-center of H ( F ) contains a translation of length . Proof.
Fix x ∈ V and let τ be as above. Consider the line L through x with labels . . . , τ − ω , τ − ω , ω , τ ω , τ ω , . . . Define t ∈ D( F ) by t ( x ) = x ω and σ ( t, y ) = τ for all y ∈ V . Then t is atranslation of length along L . Furthermore, t commutes with H ( F ) B ( x, : Indeed,let g ∈ H ( F ) B ( x, . Then ( gt )( x ) = t ( x ) = ( tg )( x ) and σ k ( gt, x ) = σ k ( g, tx ) σ k ( t, x ) = σ k ( t, x ) σ k ( g, x ) = σ k ( t, gx ) σ k ( g, x ) = σ k ( tg, x ) for all k ∈ N because σ k ( t, x ) = γ k ( τ ) ∈ Z ( F ( k ) ) and g ∈ U k +1 ( F ( k +1) ) B ( x, . (cid:3) Theorem 4.2(iv)(b).
For certain quasiprimitive F ≤ Sym(Ω) we constructa closed, non-discrete, compactly generated group H ( F ) ≤ Aut( T d ) which locallyacts like F and contains a quasi-central hyperbolic element of length .Let F ≤ Sym(Ω) be quasiprimitive. Suppose F preserves a non-trivial partition P : Ω = F i ∈ I Ω i . Further, suppose that F Ω i = { id } and that F ω i y Ω i \{ ω i } istransitive for all i ∈ I and ω i ∈ Ω i . Example 4.25.
Consider A y A /C which has blocks of size [ D : C ] = 2 andnon-trivial block stabilizers as C ∩ τ C τ − = C for all τ ∈ D given that C E D .Retain the notation of Example 3.41. Define groups F (2 k ) ≤ Aut( B d, k ) for k ∈ N inductively by F (2) = { ( a, ( a ω ) ω ) | a ∈ F, a ω ∈ C F ( a, ω ) constant w.r.t. P} and F (2( k +1)) := { ( α, ( α w ) w ) | α ∈ F (2 k ) , α w ∈ C F (2 k ) ( α, w ) , ∀ w ∈ Ω (2)0 : α w = α } . Proposition 4.26.
The groups F (2 k ) ≤ Aut( B d, k ) ( k ∈ N ) defined above satisfy:(i) Every α ∈ F (2 k ) is self-compatible in all directions from Ω (2)0 .(ii) The compatibility set C F (2 k ) ( α, w ) is non-empty for all α ∈ F (2 k ) and w ∈ Ω (2) .In particular, the group F (2 k ) satisfies (C).(iii) The compatibility set C F (2 k ) (id , w ) is non-trivial for all w ∈ Ω (2) .In particular, the group F (2 k ) does not satisfy (D). Proof.
We prove all three properties simultaneously by induction: For k = 1 , theassertion (i) holds by construction of F (2) , as do (ii) and (iii). Now assume thatall properties hold for F (2 k ) . Then the definition of F (2( k +1)) is meaningful becauseof (i) and it is a subgroup because F (2) preserves Ω (2)0 . Also, F (2( k +1)) satisfies (i)because Ω (2)0 is inversion-closed. Statements (ii) and (iii) carry over from F (2 k ) . (cid:3) Definition 4.27.
Retain the above notation. Define H ( F ) := T k ∈ N BU k ( F (2 k ) ) .Now, H ( F ) is closed as an intersection of closed sets and compactly generated by H ( F ) x for some x ∈ V and a finite generating set of BU ( { id } ) + , see Lemma 3.39.For vertices in V , the -local action is F because Γ k ( F ) ≤ F (2 k ) . For vertices in V the -local action is F + = F as Γ ( F ) ≤ F (2) . Lemma 4.28.
The group H ( F ) is non-discrete. Proof.
Let x ∈ V and n ∈ N . We construct a non-trivial element h ∈ H ( F ) whichfixes B ( x, n ) : Consider α n := id ∈ F (2 n ) : By parts (i) and (iii) of Proposition 4.5and the definition of F (2( n +1)) , there is a non-trivial element α n +1) ∈ F (2( n +1)) with π n α n +1) = α n . Applying parts (i) and (ii) of Proposition 4.26 repeatedly,we obtain non-trivial elements α k ∈ F (2 k ) for all k ≥ n + 1 with π k α k +1) = α k .Set α k := id ∈ F (2 k ) for all k ≤ n and define h ∈ Aut( T d ) x by fixing x and setting σ k ( h, x ) := α k ∈ F (2 k ) . Since F (2 l ) ≤ BΦ l ( F (2 k ) ) for all k ≤ l we conclude that h ∈ T k ∈ N BU k ( F (2 k ) ) = H ( F ) . (cid:3) Proposition 4.29.
The quasi-center of H ( F ) contains a translation of length . Proof.
Fix x ∈ V and w = ( ω , ω ) ∈ Ω (2)0 . Consider the line L through b with labels . . . , ω , ω , ω , ω , . . . Define t ∈ D( F ) by t ( x ) = x w and σ ( t, y ) = id for all y ∈ V . Then t is atranslation of length along L . Furthermore, t commutes with H ( F ) B ( x, : Indeed,let g ∈ H ( F ) B ( x, . Then gt ( x ) = t ( x ) = tg ( x ) and for all k ∈ N : σ k ( gt, x ) = σ k ( g, tx ) σ k ( t, x ) = σ k ( g, x w )= σ k ( g, x ) = σ k ( t, gx ) σ k ( g, x ) = σ k ( tg, x ) as σ l ( t, y ) = id for all l ∈ N and y ∈ V ( T d ) , and g ∈ BU k +1) ( F (2( k +1)) ) B ( b, . (cid:3) ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 33
Remark 4.30.
We argue that the construction of this section does not carry overto any primitive F ≤ Sym(Ω) and Γ( F ) ≤ F (2) ≤ Φ( F ) .First, note that Φ( F ) \ Ω (2) = Γ( F ) \ Ω (2) : For α := ( a, ( a ω ) ω ∈ Ω ) ∈ Φ( F ) and ( ω , ω ) ∈ Ω (2) we have α ( ω , ω ) = ( aω , a ω ω ) ∈ { ( aω , aF ω ω ) } ⊆ Γ( F )( ω , ω ) .We now observe the following obstruction to non-discreteness: Given any orbit Ω (2)0 ∈ Φ( F ) \ Ω (2) = F (2) \ Ω (2) , the subgroup of Φ( F ) consisting of elements whichare self-compatible in all directions from Ω (2)0 is precisely Γ( F ) :Every element of Γ( F ) is self-compatible in all directions from Ω (2) ⊇ Ω . Forthe converse, let ( a, ( a ω ) ω ) ∈ Φ( F ) be self-compatible in all directions from Ω (2)0 .Consider the equivalence relation on Ω defined by ω ∼ ω if and only if a ω = a ω .Since a ω = a ω whenever w := ( ω , ω ) ∈ Ω (2)0 , this relation is F -invariant: Since Γ( F ) ≤ Φ( F ) we have γ ( a )( ω , ω ) = ( aω , aω ) ∈ Ω (2)0 for all a ∈ F whenever ( ω , ω ) ∈ Ω (2)0 . Since F is primitive, it is the universal relation, so ( a, ( a ω ) ω ) ∈ Γ( F ) .4.2. Banks–Elder–Willis k -closures. Theorem 3.34 yields a description of the k -closures of locally transitive subgroups of Aut( T d ) which contain an involutiveinversion, and thereby a characterization of the locally transitive universal groups.Recall that the k -closure of a subgroup H ≤ Aut( T d ) is H ( k ) = { g ∈ Aut( T d ) | ∀ x ∈ V ∃ h ∈ H : g | B ( x,k ) = h | B ( x,k ) } . Combined with Corollary 3.18 the following partially answers the question for analgebraic description of a group’s k -closure raised in the last paragraph of [BEW15]. Theorem 4.31.
Let H ≤ Aut( T d ) be locally transitive and contain an involutiveinversion. Then H ( k ) = U ( l ) k ( F ( k ) ) for some labelling l of T d and F ( k ) ≤ Aut( B d,k ) . Proof.
Let l and F ( k ) ≤ Aut( B d,k ) be as in Theorem 3.34. Then H ( k ) = U ( l ) k ( F ( k ) ) :Let g ∈ U k ( F ( k ) ) and x ∈ V . Since U ( l )1 ( { id } ) ≤ H there is h ′ ∈ U ( l )1 ( { id } ) ≤ H with h ′ ( x ) = g ( x ) , and since H is k -locally action isomorphic to F ( k ) there is h ′′ ∈ H x such that σ k ( h ′′ , x ) = σ k ( g, x ) . Then h := h ′ h ′′ ∈ H satisfies g | B ( x,k ) = h | B ( x,k ) .Conversely, let g ∈ H ( k ) . Then all k -local actions of g stem from elements of H .Given that H ≤ U k ( F ( k ) ) by Theorem 3.34, we conclude g ∈ U k ( F ( k ) ) . (cid:3) Corollary 4.32.
Let H ≤ Aut( T d ) be closed, locally transitive and contain aninvolutive inversion. Then H = U ( l ) k ( F ( k ) ) for some labelling l of T d and an action F ( k ) ≤ Aut( B d,k ) if and only if H satisfies Property P k . Proof. If H = U ( l ) k ( F ( k ) ) then H satisfies Property P k by Proposition 3.7. Con-versely, if H satisfies Property P k then H = H = H ( k ) by [BEW15, Theorem 5.4]and the assertion follows from Theorem 4.31. (cid:3) Banks–Elder–Willis utilise certain subgroups of
Aut( T d ) with pairwise distinct k -closures to construct infinitely many, pairwise non-conjugate, non-discrete simplesubgroups of Aut( T d ) via Theorem 1.1 and [BEW15, Theorem 8.2]. For example, thegroup PGL(2 , Q p ) ≤ Aut( T p +1 ) qualifies by the argument in [BEW15, Section 4.1].Whereas PGL(2 , Q p ) has trivial quasi-center given that it is simple, certain groupswith non-trivial quasi-center, always have infinitely many distinct k -closures. Proposition 4.33.
Let H ≤ Aut( T d ) be closed, non-discrete, locally transitive andcontain an involutive inversion. If, in addition, H has non-trivial quasi-center then H has infinitely many distinct k -closures. Proof.
We have H ( k ) = U k ( F ( k ) ) by Theorem 4.31. Therefore, H = T k ∈ N U k ( F ( k ) ) by [BEW15, Proposition 3.4 (iii)]. If H has only finitely many distinct k -closures,the sequence ( H ( k ) ) k ∈ N of subgroups of Aut( T d ) would be eventually constant equalto, say, H ( n ) = U n ( F ( n ) ) ≥ H . However, since H is non-discrete, so is U n ( F ( n ) ) which thus has trivial quasi-center by Proposition 3.21. (cid:3) Banks–Elder–Willis ask whether the infinitely many, pairwise non-conjugate,non-discrete simple subgroups of
Aut( T d ) they construct are also pairwise non-isomorphic as topological groups. By Proposition 3.17, this is the case if said simplegroups are locally transitive with distinct point stabilizers, which can be determinedfrom the original group’s k -local actions thanks to Theorem 4.31. Theorem 4.34.
Let H ≤ Aut( T d ) be non-discrete, locally permutation isomorphicto F ≤ Sym(Ω) and contain an involutive inversion. Suppose that F is transitiveand that every non-trivial subnormal subgroup of F ω ( ω ∈ Ω) is transitive on Ω \{ ω } .If H ( k ) = H ( l ) for some k, l ∈ N then ( H ( k ) ) + k and ( H ( l ) ) + l are non-isomorphic. Proof.
In view of [BEW15, Theorem 8.2], the groups ( H ( k ) ) + k and ( H ( l ) ) + l are non-conjugate. We show that they satisfy the assumptions of Proposition 3.17 whichthen implies the assertion. It suffices to consider H ( k ) . By Theorem 4.31, we have H ( k ) = U k ( F ( k ) ) for some F ( k ) ≤ Aut( B d,k ) . By virtue of Proposition 3.10, wemay assume that F ( k ) satisfies (C). Since H is non-discrete, so is H ( k ) = U k ( F ( k ) ) .Therefore, F ( k ) does not satisfy (D), see Proposition 3.12. Hence, in view of the localaction of H and Proposition 3.31, the group π w F ( k ) T ω is non-trivial and thus transitiveby Proposition 3.30 for all w = ( ω , . . . , ω k − ) ∈ Ω ( k − and ω ∈ Ω \{ ω } . Now, let x ∈ V ( T d ) . For every ω ∈ Ω pick w = ( ω , . . . , ω k − , ω ) ∈ Ω ( k − . Let y ∈ V ( T d ) be such that x = y w . Since π w F ( k ) T ω ′ is transitive for every ω ′ ∈ Ω \{ ω } we concludethat ( H ( k ) ) + k is locally -transitive at x . Hence Proposition 3.17 applies. (cid:3) Example 4.35.
Theorem 4.34 applies to
PGL(2 , Q p ) ≤ Aut( T p +1 ) for any prime p by Lemma 4.36 below. In fact, the local action is given by PGL(2 , F p ) y P ( F p ) ,point stabilizers of which act like AGL(1 , p ) = F ∗ p ⋉ F p y F p . Retaining the notationof [BEW15, Section 4.1], an involutive inversion in PGL(2 , Q p ) is given by σ := (cid:20) p (cid:21) with σ = (cid:20) p p (cid:21) = (cid:20) (cid:21) . Indeed, σ swaps the vertices v and L p . Lemma 4.36.
Let F ≤ Sym(Ω) be -transitive. If | Ω | − is prime then everynon-trivial subnormal subgroup of F ω ( ω ∈ Ω ) acts transitively on Ω \{ ω } . Proof.
Since F ω acts transitively on Ω \{ ω } , which has prime order, F ω is primitive.So every non-trivial normal subgroup of F ω acts transitively on Ω \{ ω } . Iterate. (cid:3) Example 4.37.
The proof of Theorem 4.34 shows that the assumptions on F can be replaced with asking that ( H ( k ) ) + k be locally transitive with distinct pointstabilizers, which may be feasible to check in a given example.For instance, let F ≤ Sym(Ω) be transitive with distinct point stabilizers. Assumethat F preserves a non-trivial partition P : Ω = F i ∈ I Ω i of Ω and that it is generatedby its block stabilizers, i.e. F = h{ F Ω i | i ∈ I }i .Let p : Ω → I be such that ω ∈ Ω pω for all ω ∈ Ω . Inductively define groups F ( k ) ≤ Aut( B d,k ) by F (1) := F and F ( k +1) := Φ k ( F ( k ) , P ) , and check that(i) C F ( k ) ( α, Ω i ) is non-empty for all α ∈ F ( k ) and i ∈ I ,(ii) C F ( k ) (id , Ω i ) is non-trivial for all i ∈ I ,(iii) F ( k +1) (cid:12) Φ( F ( k ) ) , and(iv) π w F ( k ) T ω = F Ω pωk − for all ω ∈ Ω and w = ( ω , . . . , ω k − ) ∈ Ω ( k − with ω / ∈ Ω pω .In particular F ( k ) satisfies (C) but not (D) for all k ∈ N . Set H := T k ∈ N U k ( F ( k ) ) .By the above, H is non-discrete and contains both D ( F ) and U ( { id } ) . HenceTheorem 4.31 applies and we have H ( k ) = U k ( F ( k ) ) . From Item (iii), we concludethat the H ( k ) ( k ∈ N ) are pairwise distinct. Given that ( H ( k ) ) + k locally acts like F due to Item (iv), the ( H ( k ) ) + k ( k ∈ N ) are hence pairwise non-isomorphic. ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 35
A View on the Weiss Conjecture.
The Weiss conjecture states that thereare only finitely many conjugacy classes of discrete, vertex-transitive, locally prim-itive subgroups of
Aut( T d ) for a given d ∈ N ≥ . We now study the universal groupconstruction in the discrete case and thereby offer a new view on this conjecture:Under the additional assumption that each group contains an involutive inversion,it suffices to show that for every primitive F ≤ Sym(Ω) there are only finitely many e F ≤ Aut( B d,k ) ( k ∈ N ) with π e F = F and which satisfy (CD) in a minimal fashion;see Definition 4.42 and the discussion thereafter.The following consequence of Theorem 4.31 identifies certain groups relevant tothe Weiss conjecture as universal groups for local actions satisfying condition (CD). Corollary 4.38.
Let H ≤ Aut( T d ) be discrete, locally transitive and contain aninvolutive inversion. Then H = U ( l ) k ( F ( k ) ) for some k ∈ N , a labelling l of T d and F ( k ) ≤ Aut( B d,k ) satisfying (CD) which is isomorphic to the k -local action of H . Proof.
Discreteness of H implies Property P k for every k ∈ N such that stabilizersin H of balls of radius k in T d are trivial. Then apply Theorem 4.31. (cid:3) Therefore, studying the class of groups given in Corollary 4.38 reduces to study-ing subgroups F ≤ Aut( B d,k ) ( k ∈ N ) which satisfy (CD) and for which πF istransitive. By Corollary 3.15, any two conjugate such groups yield isomorphic uni-versal groups. In this sense, it suffices to examine conjugacy classes of subgroupsof Aut( B d,k ) . This can be done computationally using the description of conditions(C) and (D) developed in Section 3.2, using e.g. [GAP17]. Example 4.39.
Consider the case d = 3 . By [Tut47], [Tut59] and [DM80], there are,up to conjugacy, seven discrete, vertex-transitive and locally transitive subgroupsof Aut( T ) . We denote them by G , G , G , G , G , G and G . The subscript n determines the isomorphism class of the vertex stabilizer, whose order is · n − .A group contains an involutive inversion if and only if it has no superscript. Theminimal order of an inversion in G and G is . See also [CL89]. By Corollary 4.38,the groups G n ( n ∈ { , . . . , } ) are of the form U k ( F ) . We recover their local actionsin the following table of conjugacy class representatives of subgroups F of Aut( B , ) and Aut( B , ) which satisfy (C) and project onto a transitive subgroup of S . Thelist is complete for k = 2 , and for k = 3 in the case of (CD).Description of F k πF | F | (C) (D) i.c.c. Φ( A ) A Γ( S ) S ∆( S ) S
12 yes yes yes Π( S , sgn , { , } ) S
24 yes no no Π( S , sgn , { } ) S
24 yes no yes Φ( S ) S
48 yes no no Description of
F k π F | F | (C) (D) i.c.c. Γ (Π( S , sgn , { } )) Π( S , sgn , { } )
24 yes yes yes Σ (Π( S , sgn , { } ) , K ) Π( S , sgn , { } )
48 yes yes yesThe column labelled “i.c.c.” records whether F admits an involutive compatibilitycocycle. This can be determined in [GAP17] and is automatic in the case of (CD).The group Π( S , sgn , { } ) of Proposition 3.25 admits an involutive compatibilitycocycle z which we describe as follows: Say Ω := { , , } . Let t i ∈ Sym(Ω) be thetransposition which fixes i , and let τ i ∈ Π( S , sgn , { } ) be the element whose -localaction is t i everywhere except at b i . Then Π( S , sgn , { } ) = h τ , τ , τ i . Further, let κ i ∈ Π( S , sgn , { } ) ∩ ker π be the non-trivial element with σ ( κ i , b i ) = e . We thenhave z ( τ i , i ) = κ i − and z ( τ i , j ) = τ i κ j for all distinct i, j ∈ Ω , with cyclic notation. The kernel K is the diagonal subgroup of Z / Z · (3 − ∼ = ker π ≤ Aut( B , ) .Using the above, we conclude G = U ( A ) , G = U (Γ( S )) , G = U (∆( S )) , G = U (Γ (Π( S , sgn , { } ))) and G = U (Σ (Π( S , sgn , { } ) , K )) . Question 4.40.
Can the groups G and G be described as universal groups withprescribed local actions on edge neighbourhoods that prevent involutive inversions?The long standing Weiss conjecture [Wei78] states that there are only finitelymany conjugacy classes of discrete, vertex-transitive, locally primitive subgroupsof Aut( T d ) for a given d ∈ N ≥ . Potočnic–Spiga–Verret [PSV12] show that a per-mutation group F ≤ Sym(Ω) , for which there are only finitely many conjugacyclasses of discrete, vertex-transitive subgroups of
Aut( T d ) that locally act like F ,is necessarily semiprimitive, and conjecture the converse. Promising partial resultswere obtained in the same article as well as by Giudici–Morgan in [GM14].Corollary 4.38 suggests to restrict to discrete, locally semiprimitive subgroups of Aut( T d ) containing an involutive inversion. Conjecture 4.41.
Let F ≤ Sym(Ω) be semiprimitive. Then there are only finitelymany conjugacy classes of discrete subgroups of
Aut( T d ) which locally act like F and contain an involutive inversion.For a transitive permutation group F ≤ Sym(Ω) , let H F denote the collection ofsubgroups of Aut( T d ) which are discrete, locally act like F and contain an involutiveinversion. Then the following definition is meaningful by Corollary 4.38. Definition 4.42.
Let F ≤ Sym(Ω) be transitive. Define dim CD ( F ) := max H ∈H F min n k ∈ N (cid:12)(cid:12)(cid:12) ∃ F ( k ) ∈ Aut( B d,k ) with (CD) : H = U k ( F ( k ) ) o if the maximum exists and dim CD ( F ) = ∞ otherwise.Given Definition 4.42, Conjecture 4.41 is equivalent to asserting that dim CD ( F ) is finite whenever F ≤ Sym(Ω) is semiprimitive. The remainder of this section isdevoted to determining dim CD for certain classes of transitive permutation groups. Proposition 4.43.
Let F ≤ Sym(Ω) be transitive. Then dim CD ( F ) = 1 if andonly if F is regular. Proof. If F is regular, then dim CD ( F ) = 1 by Proposition 3.13. Conversely, if dim CD ( F ) = 1 then U (∆( F )) = U ( F ) = U (Γ( F )) . Hence Γ( F ) ∼ = ∆( F ) whichimplies that F ω is trivial for all ω ∈ Ω . That is, F is regular. (cid:3) The next proposition provides a large class of primitive groups of dimension .It relies on the following relations between various characteristic subgroups of afinite group. Recall that the socle of a finite group is the subgroup generated by itsminimal normal subgroups, which form a direct product. Lemma 4.44.
Let G be a finite group. Then the following are equivalent.(i) The socle soc( G ) has no abelian factor.(ii) The solvable radical O ∞ ( G ) is trivial.(iii) The nilpotent radical Fit( G ) is trivial. Proof. If soc( G ) has no abelian factor then O ∞ ( G ) is trivial: A non-trivial solvablenormal subgroup of G would contain a minimal solvable normal subgroup of G whichis necessarily abelian. Next, (ii) implies (iii) as every nilpotent group is solvable.Finally, if soc( G ) has an abelian factor then G contains a (minimal) normal abelian,hence nilpotent subgroup. Thus (iii) implies (i). (cid:3) Proposition 4.45.
Let F ≤ Sym(Ω) be primitive, non-regular and assume that F ω has trivial nilpotent radical for all ω ∈ Ω . Then dim CD ( F ) = 2 . ROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION 37
Proof.
Suppose that F (2) ≤ Aut( B d, ) satisfies (C) and that the sequence / / ker π / / F (2) π / / F / / is exact. Fix ω ∈ Ω . Then ker π ≤ Q ω ∈ Ω F ω ∼ = F dω . Since F (2) satisfies (C), wehave pr ω (ker π ) E F ω for all ω ∈ Ω , and since F is transitive these projections allcoincide with the same N E F ω . Now consider F (2) T ω = ker pr ω | ker π E ker π for some ω ∈ Ω . Either F (2) T ω is trivial, in which case F (2) has (CD), or F (2) T ω is non-trivial. Inthe latter case, say N ω,ω ′ := pr ω ′ F (2) T ω is non-trivial for some ω ′ ∈ Ω . Then N ω,ω ′ is subnormal in F ω as N ω,ω ′ E N E F ω and therefore has trivial nilpotent radical.The Thompson-Wielandt Theorem [Tho70], [Wie71] (cf. [BM00, Theorem 2.1.1])now implies that there is no F ( k ) ≤ Aut( B d,k ) ( k ≥ which satisfies π F ( k ) = F (2) and (CD). Thus dim CD ( F ) ≤ . Equality holds by Proposition 4.43. (cid:3) Proposition 4.45 applies to
Alt( d ) and Sym( d ) ( d ≥ ) whose point stabilizershave non-abelian simple socle Alt( d − . It also applies to primitive groups ofO’Nan-Scott type (TW) and (HS), whose point stabilizers have trivial solvableradical [DM96, Theorem 4.7B] and simple non-abelian socle [LPS88] respectively. Example 4.46.
By Example 4.39, we have dim CD ( S ) ≥ . The article [DM80]shows that in fact dim CD ( S ) = 3 .To contrast the primitive case, we show that non-trivial, imprimitive transitivewreath products have dimension at least . The proof illustrates the use of involutivecompatibility cocycles. Recall that for F ≤ Sym(Ω) and P ≤ Sym(Λ) the wreathproduct F ≀ P := F | Λ | ⋊ P admits a natural imprimitive action on Ω × Λ , given by (( a λ ) λ , σ ) · ( ω, λ ′ ) := ( a σ ( λ ′ ) ω, σλ ′ ) with block decomposition Ω × Λ = F λ ∈ Λ Ω ×{ λ } . Proposition 4.47.
Let Ω and Λ be finite sets of size at least . Furthermore, let F ≤ Sym(Ω) and P ≤ Sym(Λ) be transitive. Then dim CD ( F ≀ P ) ≥ . Proof.
We define a subgroup W ( F, P ) ≤ Aut( B | Ω × Λ | , ) which projects onto F ≀ P ,satisfies (C), does not satisfy (D) but admits an involutive compatibility cocycle.This suffices by Lemma 3.26. For λ ∈ Λ , let ι λ denote the λ -th embedding of F into F ≀ P = (cid:0) Q λ ∈ Λ F (cid:1) ⋊ P . Recall the map γ from Section 3.4.1 and consider γ λ : F → Aut( B | Ω × Λ | , ) , a ( ι λ ( a ) , (( ι λ ( a )) ( ω,λ ) , (id) ( ω,λ ′ = λ ) )) ,γ (2) λ : F → Aut( B | Ω × Λ | , ) , a (id , ((id) ( ω,λ ) , ( ι λ ( a )) ( ω,λ ′ = λ ) )) . Furthermore, let ι denote the embedding of P into F ≀ P . We define W ( F, P ) := h γ λ ( a ) , γ (2) λ ( a ) , γ ( ι ( ̺ )) | λ ∈ Λ , a ∈ F, ̺ ∈ P i . By construction, W ( F, P ) does not satisfy (D). To see that W ( F, P ) admits aninvolutive compatibility cocycle, we first determine its group structure. Considerthe subgroups V := h γ λ ( a ) | λ ∈ Λ , a ∈ F i and V := h γ (2) λ ( a ) | λ ∈ Λ , a ∈ F i .Then W ( F, P ) = h V, V , Γ( ι ( P )) i . Observe that V ∼ = F | Λ | and V ∼ = F | Λ | commute,intersect trivially and that Γ( ι ( P )) permutes the factors of each product. Hence W ( F, P ) ∼ = ( V × V ) ⋊ P ∼ = ( F | Λ | × F | Λ | ) ⋊ P. An involutive compatibility cocycle z of W ( F, P ) may now be defined by setting z ( γ λ ( a ) , ( ω, λ ′ )) := ( γ λ ( a ) λ = λ ′ γ (2) λ ( a ) λ = λ ′ , z ( γ (2) λ ( a ) , ( ω, λ ′ )) := ( γ (2) λ ( a ) λ = λ ′ γ λ ( a ) λ = λ ′ for all λ ∈ Λ , a ∈ F , and z ( γ ( ι ( ̺ )) , ( ω, λ )) := γ ( ι ( ̺ )) for all ̺ ∈ P . In fact, the map z extends to an involutive compatibility cocycle of V × V ≤ W ( F, P ) which in turnextends to an involutive compatibility cocycle of W ( F, P ) . (cid:3) References [BEW15] C. Banks, M. Elder, and G. Willis,
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