Wreath products of groups acting with bounded orbits
aa r X i v : . [ m a t h . G R ] F e b Wreath products of groups acting with boundedorbits
Paul-Henry Leemann ∗ , Grégoire SchneebergerFebruary 17, 2021 Abstract If S is a structure over (a concrete category over) metric spaces, we saythat a group G has property B S if any action on a S -space has boundedorbits. Examples of such structures include metric spaces, Hilbert spaces,CAT(0) cube complexes, connected median graphs, trees or ultra-metricspaces. The corresponding properties B S are respectively Bergman’sproperty, property FH (which, for countable groups, is equivalent to thecelebrated Kazhdan’s property (T)), property FW (both for CAT(0) cubecomplexes and for connected median graphs), property FA and uncount-able cofinality (cof = ω ).Our main result is that for a large class of structures S , the wreathproduct G ≀ X H has property B S if and only if both G and H have propertyB S and X is finite. On one hand, this encompasses in a general settingpreviously known results for properties FH and FW. On the other hand,this also applies to the Bergman’s property. Finally, we also obtain that G ≀ X H has cof = ω if and only if both G and H have cof = ω and H actson X with finitely many orbits. When working with group properties, it is natural to ask if they are stable under“natural” group operations. One such operation, of great use in geometric grouptheory, is the wreath product, which stands between the direct product andsemi-direct product of groups, see Section 2 for all the relevant definitions.A S -space is a metric space with an “additional structure” and we will saidthat a group G has property B S if every action by isometries which preservethe structure on a S -space has bounded orbits, see Definitions 2.8 and 2.9 forformal statements. We note that having one bounded orbit implies that all theorbits are bounded.In the context of properties defined by actions with bounded orbits, the firstresult concerning wreath products, due to Cherix, Martin and Valette and laterrefined by Neuhauser, concerns Kazhdan’s property (T). Theorem 1.1 ([5, 16]) . Let G and H be two discrete groups with G non-trivialand let X be a set on which H acts. The wreath product G ≀ X H has property (T)if and only if G and H have property (T) and X is finite. ∗ Supported by grant 200021_188578 of the Swiss National Fund for Scientific Research. σ -compact locally-compacttopological groups), property (T) is equivalent, by the Delorme-Guichardet’sTheorem, to property FH (every action on Hilbert space has bounded orbits),see [2, Thm. 2.12.4]. Hence, Theorem 1.1 can also be viewed, for countablegroups, as a result on property FH.The corresponding result for property FA (every action on tree has boundedorbits) is a little more convoluted and was obtained a few years later by Cornulierand Kar. Theorem 1.2 ([8]) . Let G and H be two groups with G non-trivial and let X bea set on which H acts with finitely many orbits and without fixed points. Then G ≀ X H has property FA if and only if H has property FA, G has no quotientisomorphic to Z and can not be written as a countable increasing union of propersubgroups. Finally, in a recent note, the authors proved an analogous of Theorem 1.1for property FW (every action on wall spaces has bounded orbits):
Theorem 1.3 ([14]) . Let G and H be two groups with G non-trivial and let X be a set on which H acts. Suppose that all three of G , H and G ≀ X H are finitelygenerated. Then the wreath product G ≀ X H has property FW if and only if G and H have property FW and X is finite. Since the publication of Theorem 1.3, Y. Stalder has let us know (privatecommunication) that the arguments of [14] can be adapted to replace the finitegeneration hypothesis of Theorem 1.3 by the condition that all three of G , H and G ≀ X H are at most countable. In particular, Theorem 1.3 covers the resultof A. Genevois published in [11].The above results on properties FH, FW and FA were obtained with distinctmethods even if the final results have a common flavor. In the same time, allthree properties FH, FW and FA can be characterized by the fact that any iso-metric action on a suitable metric space (respectively Hilbert space, connectedmedian graph and tree) has bounded orbits, see Definition 2.5. But more groupproperties can be characterized in terms of actions with bounded orbits. Thisis for example the case of the Bergman’s property (actions on metric spaces),property FB r (actions on reflexive Banach spaces) or of uncountable cofinality,denoted by cof = ω , (actions on ultrametric spaces).By adopting the point of view of actions with bounded orbits, we obtain aunified proof of the following result; see also Theorem 3.12 for the general (andmore technical) statement. Theorem A.
Let B S be any one of the following properties: Bergman’s prop-erty, property FB r , property FH or property FW. Let G and H be two groupswith G non-trivial and let X be a set on which H acts. Then the wreath product G ≀ X H has property B S if and only if G and H have property B S and X isfinite. With a little twist, we also obtain a similar result for groups with cof = ω : Theorem B.
Let G and H be two groups with G non-trivial and let X be a seton which H acts. Then the wreath product G ≀ X H has cof = ω if and only if G and H have cof = ω and H acts on X with finitely many orbits. G ≀ X H has property FA, then H acts on X with finitely many orbits.Combining this with Theorem 1.2 we obtain Theorem C.
Let G and H be two groups with G non-trivial and X a set onwhich H acts. Suppose that H acts on X without fixed points. Then G ≀ X H hasproperty FA if and only if H has property FA, H acts on X with finitely manyorbits, G has no quotient isomorphic to Z and can not be written as a countableincreasing union of proper subgroups. Acknowledgment
The authors are thankful to A. Genevois and T. Nag-nibeda for helpful comments and to the NSF for its support.
This section contains all the definitions, as well as some useful preliminary factsand some examples.
Let X be a set and G a group. We view L X G as the set of functions from X to G with finite support: M X G = { ϕ : X → G | ϕ ( x ) = 1 for all but finitely many x } . This is naturally a group, where multiplication is taken componentwise.If H is a group acting on X , then it naturally acts on L X G by ( h.ϕ )( x ) = ϕ ( h − .x ). This leads to the following standard definition Definition 2.1.
Let G and H be groups and X be a set on which H acts. The (restricted ) wreath product G ≀ X H is the group ( L X G ) ⋊ H .For g in G and x in X , we define the following analogs of Kronecker’s deltafunctions δ gx ( y ) := ( g y = x y = x. A prominent particular case of wreath products is of the form G ≀ H H , where H acts on itself by left multiplication. They are sometimes called standard wreathproducts or simply wreath products , while general G ≀ X H are sometimes called permutational wreath products . Best known example of wreath product is theso called lamplighter group ( Z / Z ) ≀ Z Z . Other (trivial) examples of wreathproducts are direct products G × H which correspond to wreath products overa singleton G ≀ {∗} H . There exists an unrestricted version of this product where the direct sum is remplaced bya direct product. .2 Classical actions with bounded orbits We will discuss some classical group properties which are defined by actionswith bounded orbits on various metric spaces.
Median graphs
For u and v two vertices of a connected graph G , we definethe total interval [ u, v ] as the set of vertices that lie on some shortest pathbetween u and v . A connected graph G is median if for any three vertices u , v , w , the intersection [ u, v ] ∩ [ v, w ] ∩ [ u, w ] consists of a unique vertex, denoted m ( u, v, w ). A graph is median if each of its connected components is median.For more on median graphs and spaces see [1, 4, 13]. If X and Y are both(connected) median graphs, then their cartesian product is also a (connected)median graph. Trees are the simplest examples of such graphs. The ensuingexample is important for the following. Example 2.2.
Let X be a set and let P ( X ) = 2 X be the set of all subsetsof X . Define a graph structure on P ( X ) by putting an edge between E and F ifand only if E ∆ F ) = 1, where ∆ is the symmetric difference. Therefore, thedistance between two subsets E and F is E ∆ F and the connected component of E is the set of all subsets F with E ∆ F finite. For E and F in the same connectedcomponent, [ E, F ] consists of all subsets of X that both contain E ∩ F and arecontained in E ∪ F . In particular, P ( X ) is a median graph, with m ( D, E, F )being the set of all elements belonging to at least two of D , E and F . In otherwords, m ( D, E, F ) = ( D ∩ E ) ∪ ( D ∩ F ) ∪ ( E ∩ F ).These graphs will be fundamental for us due to the following fact. Anyaction of a group G on a set X naturally extends to an action of G on P ( X ) bygraph homomorphisms: g. { x , . . . , x n } = { g.x , . . . , g.x n } . Note that the actionof G on P ( X ) may exchange the connected components. In fact, the connectedcomponent of E ⊂ X is stabilized by G if and only if E is commensurated by G ,that is if for every g ∈ G the set E ∆ gE is finite. Uncountable cofinality
Recall that a metric space (
X, d ) is ultrametric if d satisfies the strong triangular inequality: d ( x, y ) ≤ max { d ( x, z ) , d ( z, y ) } for any x , y and z in X . A group G has uncountable cofinality , or cof = ω , if everyaction on ultrametric spaces has bounded orbits. The following characterizationof groups without cof = ω is well-known and we include a proof only for the sakeof completeness. It implies in particular that a countable group has cof = ω ifand only if it is finitely generated. Lemma 2.3.
Let G be a group. Then the following are equivalent:1. G can be written as a countable increasing union of proper subgroups,2. G does not have cof = ω , i.e. there exists an ultrametric space X on which G acts with an unbounded orbit,3. There exists a G -invariant (for the action by left multiplication) ultramet-ric d on G such that G y G has an unbounded orbit. We will always assume that our connected graphs are non-empty. This is coherent withthe definition that a connected graph is a graph with exactly one connected component. roof. It is clear that the third item implies the second.Suppose that (
X, d ) is an ultrametric space on which G acts with an un-bounded orbit and let x be an element of X such that G.x is unbounded. Forany n ∈ N let H n be the subset of G defined by H n := { g ∈ G | d ( x , g.x ) ≤ n } . It is clear that G is the increasing union of the (countably many) H n . On theother hand, the H n are subgroups of G . Indeed, H n is trivially closed undertaking the inverse, and is also closed under taking products since d ( x , gh.x ) ≤ max { d ( x , g.x ) , d ( g.x , gh.x ) } = max { d ( x , g.x ) , d ( x , h.x ) } . As G.x is un-bounded, they are proper subgroups.Finally, suppose that G = S n ∈ N H n where the H n form an increasing se-quence of proper subgroups. It is always possible to suppose that H = { } .Define d on G by d ( g, h ) := min { n | g − h ∈ H n } . One easily verifies that d is a G -invariant ultrametric. Moreover, the orbit of 1 contains all of G and is henceunbounded.A slight variation of the above lemma gives us Example 2.4.
Let ( G i ) i ≥ be non-trivial groups and let G := L i ≥ G i be theirdirect sum. Then d ∞ ( f, g ) := max { i | f ( i ) = g ( i ) } is an ultrametric on G whichis G -invariant for the action of G on itself by left multiplication. Some classical group properties
We now discuss the bounded orbits prop-erties introduced above for actions on various classes of metric spaces.
Definition 2.5.
Let G be a group. It is said to have• Bergman’s property if any action on a metric space has bounded orbits,• property FB r if any action on a reflexive real Banach space has boundedorbits,• property FH if any action on a real Hilbert space has bounded orbits,• property FW if any action on a connected median graph has boundedorbits,• property FA if any action on a tree has bounded orbits,• cof = ω if any action on an ultrametric space has bounded orbits.In the above, actions are supposed to preserve the structure. In particular,actions on (ultra)metric spaces are by isometries, actions on graphs (includingtrees) are by graph isomorphisms and actions on Hilbert spaces are by scalarproduct preserving isometries.The names FB r , FH, FW and FA come from the fact that these propertiesadmit a description in terms of (and were fist studied in the context of) theexistence of a Fixed point for actions on reflexive Banach spaces, on Hilbertspaces, on spaces with Walls (or equivalently on CAT(0) cube complexes) andon trees ( Arbres in french). For a survey on property FB r , see [17] and thereferences therein. 5or countable groups (and more generally for σ -compact locally compactgroups), property FH is equivalent to the celebrated Kazhdan’s property (T)by the Delorme-Guichardet theorem, but this is not true in general. Indeedin [9], Cornulier constructed an uncountable discrete group G with Bergman’sproperty which, as we will see just below, implies property FH. Such a groupcannot have property (T) as, for discrete groups, it implies finite generation.A classical result of the Bass-Serre theory of groups acting on trees [18], isthat a group G has property FA if and only if it satisfies the following threeconditions: G has cof = ω , G has no quotient isomorphic to Z and G is not anon-trivial amalgam. In view of this characterization, Theorem 1.2 says thatproperty FA almost behave well under wreath products. Proposition 2.6.
There are the following implications between the propertiesof Definition 2.5:
Bergman’s property = ⇒ FB r = ⇒ FH = ⇒ FW = ⇒ FA = ⇒ cof = ω. (†) Moreover, except maybe for the leftmost one, all implications are strict.Proof.
The implications Bergman’s property = ⇒ FB r = ⇒ FH and FW = ⇒ FA trivially follow from the fact that Hilbert spaces are reflexive Banach spaces,which are themselves metric spaces and that trees are connected median graphs.The implication FA = ⇒ cof = ω is due to Serre [18]: if G is an increasingunion of subgroups G i , then F G/G i admits a tree structure by joining any gG i ∈ G/G i to gG i +1 ∈ G/G i +1 . The action of G by multiplication on F G/G i is by graph isomorphisms and with unbounded orbits. Finally, the implicationFH = ⇒ FW follows from the fact that a group G has property FW if andonly if any action on a real Hilbert space which preserves integral points hasbounded orbits [6].On the other hand, here are some examples for the strictness of the implica-tions of (†). Countable groups with property FB r are finite by [3], while infinitefinitely generated groups with property (T), e.g. SL ( Z ), have property FH.The group SL ( Z [ √ G is a non-trivial finite group and H is an infinite group with property FA, then G ≀ H H has property FA by Theorem C, but does not have property FW by Theorem A.Finally, Z has cof = ω , while it acts by translations and with unbounded orbitson the infinite 2-regular tree.It is possible to consider relative versions of the properties appearing inDefinition 2.5. If G is a group and H a subgroup of G , we say that the pair( G, H ) has relative property B S if for every G action on a S -space, the H orbitsare bounded. A group G has property B S if and only if for every subgroup H the pair ( G, H ) has relative property B S , and if and only if for every overgroup L the pair ( L, G ) has relative property B S . It is possible to define other properties in the spirit of Definition 2.5 for any“additional structure on metric spaces”. In order to give a uniform treatment ofall of them, we will use the notion of quasipseudo-metric spaces and the languageof category theory. A reader not familiar with category theory and interested6nly in one specific structure may forget all these general considerations andonly verify that the arguments of Section 3 apply for their favorite structure.
Definition 2.7. A quasipseudo-metric space is a set X with a map d : X × X → R ≥ , called a quasipseudo-distance , such that1. d ( x, x ) = 0 for all x ∈ X ,2. d ( x, z ) ≤ d ( x, y ) + d ( y, z ).If moreover d ( x, y ) = d ( y, x ) and d ( x, y ) = 0 for x = y , the map d is a distance and ( X, d ) is a metric space. On the other hand, an ultra-quasipseudo-metric space is a quasipseudo-metric space (
X, d ) such that d satisfies the strongtriangular inequality. A morphism (or short map ) between two quasipseudo-metric spaces ( X , d ) and ( X , d ) is a distance non-increasing map f : X → X , that is d ( f ( x ) , f ( y )) ≤ d ( x, y ) for any x and y in X . If f is bijectiveand distance preserving, then it is an isomorphism (or isometry ). Quasipseudo-metric spaces with short maps form a category QPMet of which the categoryof metric spaces (with short maps)
Met is a full subcategory.If (
X, d ) is a quasipseudo-metric space, we have a natural notion of the diameter of a subset Y ⊂ X with value in [0 , ∞ ], defined by diam( Y ) :=sup { d ( x, y ) | x, y ∈ Y } . Definition 2.8. An additional structure on quasipseudo-metric spaces , or a qp-metric structure for short, is a concret category ( S , F S ) over QPMet . Thatis, it is a category S together with a faithful functor F S : S → QPMet . Theobjects of S are called S -spaces and the morphisms S -morphisms .A G -action on a S -space X is simply an homomorphism α : G → Aut S ( X ).It has bounded orbits if F S ◦ α : G → Aut
QPMet ( X ) has bounded orbits.In practice, we will often simply write S for the pair ( S , F S ). Since QPMet itself is concrete, that is we have a faithful functor F : QPMet → Set , thecategory S is also concrete (via F ◦ F S ) and its objects can be thoughts as setswith “extra structure”.In practice, a lot of examples of concrete categories over QPMet factorthrough the category
Met . Obvious examples of concrete categories over
Met include metric spaces and ultrametric spaces (with short maps). Hilbert spaces,Banach spaces and more generally (semi-) normed spaces are also concrete over
Met if we restrict ourself to morphisms that do not increase the distance (thatis such that h f ( x ) | f ( y ) i ≤ h x | y i , respectively k f ( x ) k ≤ k x k ). In particular,for us isomorphisms of Hilbert and Banach spaces will always be isometries.For connected graphs (and hence for connected median graphs and for trees),one looks at the category Graph where objects are connected simple graphs G =( V, E ) and where a morphism f : ( V, E ) → ( V ′ , E ′ ) is a function between thevertex sets such that if ( x, y ) is an edge then either f ( x ) = f ( y ) or ( f ( x ) , f ( y ))is an edge. The functor F S : Graph → Met sends a connected graph to itsvertex set together with the graph distance: d ( x, y ) is the minimum number ofedges on a path between x and y . We can hence identify Graph with the fullsubcategory of
Met consisting of metric spaces (
X, d ) such that d has valuesin N .We can now formally define the group property B S as:7 efinition 2.9. Let ( S , F S ) be a qp-metric structure. A group G has propertyB S if every G -action on a S -space has bounded orbits. A pair ( G, H ) of a groupand a subgroup has relative property B S if for every G -action on a S -space, the H orbits are bounded.All the properties of Definition 2.5 are of the form B S . Examples of otherinteresting properties include R -trees (also called real trees ) or (some specificsubclass of) Banach spaces. The property F R of having bounded actions on R -trees is known to be strictly stronger than FA [15]. Theorem C holds wheneverproperty FA is replaced by property F R , with a similar proof.Another interesting example of a property of the form B S is the fact to haveno quotient isomorphic to Z , see Example 2.10. The main interest for us of thisexample is that property FA is the conjonction of three properties, two of them(cof = ω and having no quotient isomorphic to Z ) still being of the form B S . Example 2.10.
Let Z n be the integer lattice of dimension n , that is Z n isthe Cayley graph of Z n for the standard generating set. Then every infinitesubgroup of Aut( Z n ) projects onto Z . Hence, we obtain that a group G has noquotient isomorphic to Z if and only if every G action on Z has bounded orbits,if and only if every G action on Z n has bounded orbits. Let us denote by B Z this property.It follows from Bass-Serre theory that FA implies B Z . This implication isstrict as demonstrated by Q . In fact, the counterexample Q shows that B Z does not implies cof = ω . By looking at Z , we see that cof = ω does not impliesB Z neither.An example of an uninteresting property B S is given by taking S to bethe category of metric spaces of bounded diameter (together with short maps).Indeed, in this case, any group has B S . On the opposite, if S is the categoryof extended quasipseudo-metric spaces ( d takes values in R ∪ {∞} ), only thetrivial group has B S .The category QPMet has the advantage (over
Met ) of behaving more nicelywith respects to categorical constructions. However, we have
Lemma 2.11.
A group G has Bergman’s property (respectively cof = ω ) if andonly if any G action on a quasipseudo-metric (respectively ultra-quasipseudo-metric) space has bounded orbits.Proof. One direction is trivial.For the other direction, let (
X, d ) be a quasipseudo-metric space on which G acts by isometries. Let d ′ ( x, y ) := ( d ( x, y ) + d ( y, x )) be the symmetrizationof d . Then the action of G on X is by d ′ -isometries and a subset Y ⊂ X isbounded for d if and only if it is bounded for d ′ . Finally, let ˜ X := X/ ∼ be thequotient of X for the relation x ∼ y if d ′ ( x, y ) = 0 and let ˜ d be the quotient of d ′ . Then ( ˜ X, ˜ d ) is a metric space, the action of G passes to the quotient and G.x is d ′ bounded (hence d bounded) if and only if G. [ x ] is ˜ d bounded. Finally,if d satisfies the strong triangular inequality, then so do d ′ and ˜ d .On the other hand, the following result is perhaps more surprising. Lemma 2.12.
A group G has Bergman’s property if and only if any G actionon a connected graph has bounded orbits. roof. The left-to-right implication is clear.For the other direction, we will use the following characterization of Bergman’sproperty due to Cornulier [9]. A group G has Bergman’s property if and only ifit has cof = ω and for every generating set T of G the Calyey graph Cayl( G ; T )is bounded.Suppose that any G action on a connected graph has bounded orbits. Then G has property FW and hence cof = ω . On the other hand, for every generating set T , the group G acts transitively on Cayl( G ; T ), which implies that the latter isbounded. By the above characterization, we obtain that G has property SB.While we will be able to obtain some results for a general qp-metric struc-ture S , we will sometimes need to restrict ourselves to structures with a suitablenotion of cartesian product. Definition 2.13.
A qp-metric structure ( S , F S ) has cartesian powers if for any S -space X and any integer n , there exists a S -object, called the n th cartesianpower of X and written X n , such that:1. X n is compatible with the cartesian product of sets. That is F ◦ F S ( X n )is the set cartesian power.2. If E ⊂ X is unbounded, then the diagonal diag( E ) ⊂ X n is unbounded.3. Aut S ( X ) n ⋊ Sym( n ) is a subgroup of Aut S ( X n ).For (quasipseudo-/ultra-) metric spaces, the categorical product (correspond-ing to the metric d ∞ = max { d X , d Y } ) works fine, but any product metric ofthe form d p = ( d pX + d pY ) p for p ∈ [1 , ∞ ] works as well. For Hilbert and Banachspaces, we take the usual cartesian product (which is also the categorial prod-uct), which corresponds to the metric d = p d X + d Y . For connected mediangraphs, the usual cartesian product (which is not the categorical product! )with d = d X + d Y works well. On the other hand, trees do not have cartesianpowers. Remark 2.14.
In view of Definition 2.9 and 2.13, the reader might ask why weare working in
QPMet instead of
Born , the category of bornological spaces to-gether with bounded maps. The reason behind this is the forthcoming Lemma 3.2and its corollaries, which fail for general bornological spaces.
Groups acting with fixed point on S-spaces
Some of the properties thatare of interest for us have been historically defined via the existence of a fixedpoint for some action. More generally, we say that a group G has property F S if any G action on a S -space has a fixed point.Since our actions are by isometries, property F S implies property B S . Theother implication holds as soon as we have a suitable notion of the center ofa (non-empty) bounded subset X . For a large class of metric spaces, this isprovided by the following result of Bruhat and Tits: Proposition 2.15 ([10, Chapter 3.b]) . Let ( X, d ) be a complete metric spacesuch that the following two conditions are satisfied: The categorial product in
Graph is the strong product. . For all x and y in X , there exists a unique m ∈ X (the middle of [ x, y ] )such that d ( x, m ) = d ( y, m ) = d ( x, y ) ;2. For all x , y and z in X , if m is the middle of [ y, z ] we have the median’sinequality d ( x, m ) + d ( y, z ) ≤ d ( x, y ) + d ( x, z ) .Then if G is a group acting by isometries on X with a bounded orbit, it has afixed point. Exemples of complete metric spaces satisfying Proposition 2.15 include amongothers: Hilbert spaces, Bruhat-Tits Buildings, Hadamard spaces (i.e. completeCAT(0) spaces), trees and R -trees; with the caveat that for ( R -)trees, the fixedpoint is either a vertex or the middle of an edge. See [10, Chapter 3.b] and thereferences therein for more on this subject. On the other hand, [2, Lemma 2.2.7]gives a simple proof of the existence of a center for bounded subsets of Hilbertspaces, and more generally of reflexive Banach spaces.For action on (ultra)-metric spaces or on connected median graphs, F S isstrictly stronger than B S . Indeed, this trivially follows from the action byrotation of C on the square graph. However, by [6, 12] if a group G acts on aconnected median graph with a bounded orbit, then it has a finite orbit. Remark 2.16.
We conclude this section by a remark on a variation of Def-inition 2.5. One might wonder what happens if in Definition 2.5 we replacethe requirement of having bounded orbits by having uniformly bounded orbits.It turns out that this is rather uninteresting, as a group G is trivial if andonly if any G -action on a metric space (respectively on an Hilbert space, on aconnected median graph, on a tree or on an ultrametric space) has uniformlybounded orbits. Indeed, if G is non-trivial, then, for the action of G on theHilbert space ℓ ( G ) the orbit of n · δ g has diameter n √
2. For a tree (and hencealso for a connected median graph), one may look at the tree T obtained bytaking a root r on which we glue an infinite ray for each element of G . Then G naturally acts on T by permuting the rays. The orbits for this action are the L n = { v | d ( v, r ) = n } which have diameter 2 n . Finally, it is possible to put anultradistance on the vertices of T by d ∞ ( x, y ) := max { d ( x, r ) , d ( y, r ) } if x = y .Then the orbits are still the L n , but this time with diameter n . Throughout this section, S will denote a qp-metric structure and B S the groupproperty “every action on a S -space has bounded orbits”. Heuristically, a S -space is a (quasipseudo-) metric space with an additional structure (as for ex-ample an Hilbert space). For a precise definition, see Definition 2.9.We begin this section with two easy but useful results. Lemma 3.1.
Let G be a group and H be a quotient. If G has property B S ,then so has H .Proof. We have H ∼ = G/N . If H acts on some S -space X with an unboundedorbit, then the G action on X defined by g.x := gN.x has also an unboundedorbit. 10 emma 3.2. Let G be a group and A an B be two subgroups such that G = AB .If both ( G, A ) and ( G, B ) have relative property B S , then G has property B S .Proof. Let X be a S -space on which G acts and let x be an element of X . Let D be the diameter of A.x and D the diameter of B.x . By assumption, theyboth are finite. Since A acts by isometries, all the a.Bx have diameter D . Let y be an element of G.x . There exists a ∈ A such that y belongs to a.Bx . Since1 belongs to B , y is at distance at most D of a.x and hence at distance at most D + D of x . Therefore, the diameter of G.x is finite.By combining Lemmas 3.1 and 3.2, we obtain the following three corollarieson direct, semi-direct and wreath products.
Corollary 3.3.
Let G and H be two groups. Then G × H has property B S ifand only if both G and H have property B S . Corollary 3.4.
Let N ⋊ H be a semidirect product. Then1. If N ⋊ H has property B S , then so has H .2. If both N and H have property B S , then N ⋊ H also has property B S . Corollary 3.5.
Let G and H be two groups and X a set on which H acts.Then,1. If G ≀ X H has property B S , then so has H ,2. If both G and H have property B S and X is finite, then G ≀ X H hasproperty B S . When S has a suitable notion of quotients (by a group of isometries), it ispossible to obtain a strong version of Lemmas 3.1 and 3.2. Here is the corre-sponding result for Bergman’s property and cof = ω . Proposition 3.6.
Let B S be either Bergman’s property or the property cof = ω .Let → N → G → H → be a group extension. Then G has property B S ifand only if H has property B S and the pair ( G, N ) has the relative B S property.Proof. One direction is simply Lemma 3.1 and the definition of relative prop-erty B S .On the other hand, let ( X, d ) be a quasipseudo-metric space on which G actsby isometries and let x be an element of X . Let { g i | i ∈ I } be a transversalfor N , that is H ∼ = { g i N } . By assumption, N.x is bounded of diameter D and for any i ∈ I the subset g i N.x of X has also diameter D . Since N is a subgroup of isometries of X , the map d ′ : X/N × X/N → R defined by d ′ ([ x ] , [ y ]) := inf { d ( x ′ , y ′ ) | x ′ ∈ N.x, y ′ ∈ N.y } is the quotient quasipseudo-distance on X/N . Indeed, while the map d ′ might not satisfies the triangleinequality for a generic quotient X/ ∼ , this is the case if the quotient is by asubgroup of isometries; details are left to the reader. Moreover, if d satisfies thestrong triangle inequality, then so does d ′ . The quotient action of H ∼ = G/N on X/N is by isometries and the diameter of
H.xN is bounded, say by D . Inparticular, for any i and j in I , the distance between the subsets g i N.x and g j N.x of X is bounded by D . Since this distance is an infimum, there existsactual elements of g i N.x and g j N.x at distance less than D + 1. Altogether,we obtain that any y in G.x is at distance at most D + D + 1 of x . Hence,the orbit G.x is bounded. 11ince the square graph, which is not median, is a quotient of the 2-regularinfinite tree by a subgroup of isometries, the proof of Proposition 3.6 does notcarry over for properties FW and FA. Similarly, the quotient of R by the ac-tion of Z / Z given by x
7→ − x is not an Hilbert space and hence the proofof Proposition 3.6 does not apply to property FH. However, the statement ofProposition 3.6 (stability under extension) remains true for properties FH, FWand FA. For property FH this is an exercice using the fixed-point definition, forproperty FW and FA, see [6] and [18].We now state a result on infinite direct sums. Lemma 3.7.
Suppose that B S implies cof = ω . Let ( G x ) x ∈ X be non-trivialgroups. Then1. L x ∈ X G x has property B S if and only if all the G x have property B S and X is finite,2. If G ≀ X H has property B S , then H acts on X with finitely many orbits. It is of course possible to prove Lemma 3.7 using the characterization ofcof = ω in terms of subgroups. However, we find enlightening to prove it usingthe characterization in terms of actions on ultrametric spaces. Proof of Lemma 3.7.
One direction of the first assertion is simply Corollary 3.3.For the other direction, if L x ∈ X G x has property B S then all its quotients, andhence all the G x , have property B S . On the other hand, if X is infinite, thereexists a countable subset Y ⊂ X . Let Z := X \ Y , thus we have X = Y ⊔ Z .We can decompose the direct sum as L X G = ( L Y G ) × ( L Z G ) and then,by Corollary 3.3, if L Y G does not have cof = ω , then neither does L X G .So let G := L i ≥ G i and for each i , choose g i = 1 in G i . Let d ∞ ( f, g ) :=max { i | f ( i ) = g ( i ) } be the G -invariant ultrametric of Example 2.4. Then forevery integer n , the orbit G. G contains { g , . . . , g n , , . . . } which is at distance n of 1 G for d ∞ if the g i are not equal to 1. In particular, an infinite direct sumof non-trivial groups does not have cof = ω , nor does it have B S .The second assertion is a simple variation on the first. Indeed, we have G ≀ X H ∼ = ( M Y ∈ X/H L Y ) ⋊ H with L Y ∼ = M y ∈ Y G y , where X/H is the set of H -orbits. The important fact for us is that H fixesthe decomposition into L Y factors: for all Y we have H.L Y = L Y . Up toregrouping some of the L Y together we hence have G ≀ X H ∼ = (cid:0)L i ≥ L i (cid:1) ⋊ H with H.L i = L i for all i . Now, we have an ultradistance d ∞ on L := L i ≥ L i as above and we can put the discrete distance d on H . Then d ′∞ = max { d ∞ , d } is an ultradistance on (cid:0)L i ≥ L i (cid:1) ⋊ H , which is (cid:0)L i ≥ L i (cid:1) ⋊ H -invariant (forthe action by left multiplication). From a practical point of view, we have d ′∞ (cid:0) ( ϕ, h ) , ( ϕ ′ , h ′ ) (cid:1) := max { i | ϕ ( i ) = ϕ ′ ( i ) } if ϕ = ϕ ′ and d ′∞ (cid:0) ( ϕ, h ) , ( ϕ, h ′ ) = 1if h = h ′ . Since the action of L on itself has an unbounded orbit for d ∞ , theaction of (cid:0)L i ≥ L i (cid:1) ⋊ H on itself has an unbounded orbit for d ′∞ .While the statement (and the proof) of Lemma 3.7 is expressed in terms ofcof = ω , it is also possible to state it and prove it for a qp-metric structure S without a priori knowing if B S is stronger than cof = ω . The main idea is12o find a “natural” S -space on which G = L i ≥ G i acts. For example, forHilbert spaces, one can take L i ≥ ℓ ( G i ). For connected median graphs, onetakes the connected component of { G , G , . . . } in P ( F i ≥ G i ). For trees, itis possible to put a forest structure on P ( F i ≥ G i ) in the following way. For E ∈ P ( F i ≥ G i ), and for each i such that E ∩ G j is empty for all j ≤ i , addan edge from E to E ∪ { g } for each g ∈ G i . The graph obtained this way is a G -invariant subforest of the median graph on P ( F i ≥ G i ). Lemma 3.8.
Suppose that B S implies FW. Let G and H be two groups with G non-trivial and let X be a set on which H acts. If G ≀ X H has B S , then X isfinite.Proof. We will prove that if X is infinite, then G ≀ X H does not have propertyFW. Suppose that X is infinite. The group L X G acts coordinatewise on F X G :the group G x acting by left multiplication on G x and trivially on G y for y = x .On the other hand, H acts on F X G by permutation of the factors. Altogetherwe have an action of G ≀ X H on F X G and hence on the median graph P ( F X G ).Let := S x ∈ X G be the subset of P ( F X G ) consisting of the identity elementsof all the copies of G . Since every element of L X G has only a finite number ofnon-trivial coordinates, the action of G ≀ X H preserves the connected componentsof (and in fact every connected component of P ( F X G )).Let I = { i , i , . . . } be a countable subset of X and for every i ∈ I , choosea non-trivial g i ∈ G i . Then the orbit of the vertex contains the point { g i , . . . , g i n } ∪ (cid:0)S j>n G ij (cid:1) ∪ (cid:0)S x/ ∈ I G x (cid:1) which is at distance 2 n of . There-fore, the action of G ≀ X H on the connected component of has an unboundedorbit and then G ≀ X H does not have property FW.Once again, given a suitable S , it is sometimes possible to give a direct proofof Lemma 3.8. For example, for Hilbert spaces one can take L X ℓ ( G ) with L X G acting coordinatewise and H by permutations. On the other hand, boththe forest structure on P ( F X G ) and the ultrametric structure on L X G are ingeneral not invariant under the natural action of H by permutations.In fact, it follows from Theorems 1.2 and B that in the assumptions ofLemma 3.8 it is not possible to replace property FW by property FA or bycof = ω . Remark 3.9.
A reader familiar with wreath products might have recognizedthat we used the primitive action of the wreath product in the proof of Lemma 3.8.Indeed, G acts on itself by left multiplication. It hence acts on the set G ′ := G ⊔ { ε } by fixing ε , and we have the primitive action of G ≀ X H on G ′ X .Now, the set F X G naturally embeds as the subset of G ′ X consisting of allfunctions ϕ : X → G ′ such that ϕ ( x ) = ε for all but one x ∈ X . This subset is G ≀ X H invariant, which gives us the desired action of G ≀ X H on F X G .We now turn our attention to properties that behave well under cartesianproducts in the sense of Definition 2.13.We first describe the comportement of property B S under finite index sub-groups. Lemma 3.10.
Let G be a group and let H be a finite index subgroup.1. If H has property B S , then so has G , . If S has cartesian powers and G has property B S , then H has propertyB S .Proof. Suppose that G does not have B S and let X be a S -space on which G acts with an unbounded orbit O . Then H acts on X and O is a union of atmost [ G : H ] orbits. This directly implies that H has an unbounded orbit andtherefore does not have B S .On the other hand, suppose that H ≤ G is a finite index subgroup of Gwithout property B S . Let α : H y X be an action of H on a S -space ( X, d X )such that there is an unbounded orbit O . Similarly to the classical theoryof representations of finite groups, we have the induced action Ind GH ( α ) : G y X G/H on the set X G/H . Since H has finite index, X G/H is a S -space andthe action is by S -automorphisms. On the other hand, the subgroup H ≤ G acts diagonally on X G/H , which implies that diag( O ) is contained in a G -orbit.Since diag( O ) is unbounded, G does not have property B S .For readers that are not familiar with representations of finite groups, hereis the above argument in more details. Let ( f i ) ni =1 be a transversal for G/H .The natural action of G on G/H gives rise to an action of G on { , . . . , n } .Hence, for any g in G and i in { , . . . , n } there exists a unique h g,i in H suchthat gf i = f g.i h g,i . That is, h g,i = f − g.i gf i . We then define g. ( x , . . . , x n ) :=( h g,g − . .x g − . , . . . , h g,g − .n .x g − .n ). This is indeed an action by S -automor-phisms on X G/H by Condition 3 of Definition 2.13. Moreover, every element h ∈ H acts diagonally by h. ( x , . . . , x n ) = ( h.x , . . . , h.x n ). In particular, this G action has an unbounded orbit.We now prove one last lemma that will be necessary fo the proof of Theo-rem A. Lemma 3.11.
Suppose that S has cartesian powers. If X is finite and G ≀ X H has property B S , then G has property B S .Proof. Suppose that G does not have B S and let ( Y, d Y ) be a S -space on which G acts with an unbounded orbit G.y . Then ( Y X , d ) is a S -space and we havethe primitive action of the wreath product G ≀ X H on Y X : (cid:0) ( ϕ, h ) .ψ (cid:1) ( x ) = ϕ ( h − .x ) .ψ ( h − .x ) . By Condition 3 of Definition 2.13, this action is by S -automorphisms. Theorbit G.y embeds diagonally and hence diag(
G.y ) is an unbounded subset ofsome G ≀ X H -orbit, which implies that G ≀ X H does not have property B S .By combining Corollary 3.5 and Lemmas 3.8 and 3.11 we obtain the followingresult which implies Theorem A. Theorem 3.12.
Suppose that S has cartesian powers and is such that B S im-plies FW. Let G and H be two groups with G non-trivial and let X be a set onwhich H acts. Then the wreath product G ≀ X H has property B S if and only if G and H have property B S and X is finite. We now proceed to prove Theorem B. As for Lemma 3.7, it is also possible toprove it using the characterization of cof = ω in terms of subgroups, but we willonly give a proof using the characterization in terms of actions on ultrametricspaces. 14 roof of Theorem B. By Corollary 3.5 and Lemma 3.7 we already know that if G ≀ X H has cof = ω , then H has cof = ω and it acts on X with finitely manyorbits. We will now prove that if G ≀ X H has cof = ω so does G . Let us supposethat G has countable cofinality. By Lemma 2.3, there exists an ultrametric d on G such that the action of G on itself by left multiplication has an unboundedorbit. But then we have the primitive action of the wreath product G ≀ X H on G X ∼ = Q X G , which preserves L X G . It is easy to check that the map d ∞ : L X G × L X G → R defined by d ∞ ( ψ , ψ ) := max { d (cid:0) ψ ( x ) , ψ ( x ) (cid:1) | x ∈ X } is a G ≀ X H -invariant ultrametric. Finally, let h ∈ G be an element ofunbounded G -orbit for d and let x be any element of X . Then for any g in G we have ( δ gx , .δ hx = δ ghx and hence d ∞ ( δ hx , δ ghx ) = d ( h, gh ) is unbounded.Suppose now that both G and H have cof = ω and that H acts on X withfinitely many orbits. We want to prove that G ≀ X H has cof = ω .Let ( Y, d ) be an ultrametric space on which G ≀ X H acts. Then H and allthe G x act on Y with bounded orbits. Let O , . . . , O n be the H -orbits on X and for each 1 ≤ i ≤ n choose an element x i in O i . Let y be any element of Y .Then H.y has finite diameter D while G x i .y has finite diameter D i . For any x ∈ X , there exists 1 ≤ i ≤ n and h ∈ H such that x = h.x i . We have d (cid:0) ( δ gx i , h − ) .y, y (cid:1) ≤ max { d (cid:0) ( δ gx i , h − ) .y, ( δ gx i , .y (cid:1) , d (cid:0) ( δ gx i , .y, y (cid:1) } = max { d (cid:0) (1 , h − ) .y, y (cid:1) , d (cid:0) ( δ gx i , .y, y (cid:1) }≤ max { D , D i } , which implies that the diameter of G x i h − .y is bounded by max { D , D i } . But G x i h − .y has the same diameter as hG x i h − .y = G h.x i .y = G x .y .On the other hand, the diameter of L X G.y is bounded by the supremumof the diameters of the G x i .y , and hence bounded by max { D , D , . . . , D n } .Finally, for ( ϕ, h ) in G ≀ Y H we have d (cid:0) y, ( ϕ, h ) .y (cid:1) ≤ max { d (cid:0) y, ( ϕ, .y (cid:1) , d (cid:0) ( ϕ, .y, ( ϕ, h ) .y (cid:1) } = max { d (cid:0) y, ( ϕ, .y (cid:1) , d (cid:0) y, (1 , h ) .y (cid:1) }≤ max { max { D , D , . . . , D n } , D } . That is, the diameter of G ≀ Y H.z is itself bounded by max { D , D , . . . , D n } ,which finishes the proof.While the fact that trees do not have cartesian powers is an obstacle to ourmethods, we still have a weak version of Theorem 3.12 for property FA. Beforestating it, remind that we already know, by Theorem B, the behavior of cof = ω under wreath products. On the other hand, we have the following result (whichalso admits a direct algebraic proof) Lemma 3.13.
Suppose that H acts on X with finitely many orbits. Then thegroup G ≀ X H has no quotient isomorphic to Z if and only if both G and H haveno quotient isomorphic to Z .Proof. Remind that a group K has no quotient isomorphic to Z if and onlyif K ab has no quotient isomorphic to Z . The desired result then follows from( G ≀ X H ) ab ∼ = L X/H ( G ab ) × H ab and Corollary 3.3.By Corollary 3.5, Theorem B and Lemma 3.13, we directly obtain the fol-lowing partial version of Theorem 1.2.15 roposition 3.14. Let G and H be two groups with G non-trivial and X a seton which H acts. Then1. If G ≀ X H has property FA, then H has property FA, H acts on X withfinitely many orbits, G has no quotient isomorphic to Z and G has cof = ω ,2. If both G and H have no quotient isomorphic to Z , have cof = ω and H acts on X with finitely many orbits, then G ≀ X H has no quotient isomor-phic to Z and has cof = ω ,3. If both G and H have property FA and X is finite, then G ≀ X H hasproperty FA. Moreover, by using Lemma 3.7 we can get ride of the “finitely many orbits”hypothesis in Theorem 1.2 in order to obtain Theorem C.Observe that our statement of Theorem 1.2 differs of the original statementof [8]. Indeed, where we ask G to have cof = ω and no quotient isomorphic to Z ,the authors of [8] ask G to have cof = ω and finite abelianization. However, thesetwo sets of conditions are equivalent. One implication is trivial, as finite abeliangroups do not projects onto Z . The other implication follows, for countableabelian groups, from the structure theorem of finitely generated abelian groups.For the general case, Y. Cornulier kindly reminded us that any infinite abeliangroup as a countable quotient. References [1] Hans-Jürgen Bandelt and Victor Chepoi. Metric graph theory and geome-try: a survey. In
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