SSURGERY ON
Aut ( F ) SYLVAIN BARRÉ AND MIKAËL PICHOT
Abstract.
We study a geometric construction of certain finite index sub-groups of
Aut ( F ) . We recall that
Aut ( F ) admits an isometric properly discontinuous action withcompact quotient on a CAT(0) complex X , called the Brady complex, which wasintroduced in [4]In §1, we show that Aut ( F ) can be presented (virtually) in a very simple mannerfrom a labelling of a flat torus. Starting from a torus of size × n , for some fixedinteger n ≥ (we shall discuss the case n = in details), we associate to it, viaa “pinching and (systolic) filling” construction, a 2-complex B n , with fundamentalgroup a group G n which is of finite index in Aut ( F ) . The universal cover X n of B n is a CAT(0) space. We show in §1 that the X n ’s are pairwise isometric for every n ≥ , and that X and X n are locally isometric, in the sense that their vertex linksare pairwise isometric (Lemma 1.3) for every n ≥ . This implies, by the resultbelow, that X n is isometric to X for every n .In §2, we prove a geometric rigidity theorem for the Brady complex. Roughlyspeaking, the result states that X is the “free complex” on one (any) of its face,among the complexes locally isomorphic to X (see Th. 2.3 for a precise statement).This seems to be a rather special property of X , which is not very often satisfiedamong the 2-complexes we have studied.Theorem 2.3 implies that every CAT(0) 2-complex locally isometric to X isisometric to X . The notion of local isomorphism in this statement is slightly morerestrictive than requiring the existence of an abstract isometry between the linksshown in Lemma 1.3: the two complexes must be of the same (local) type (see §2).The additional conditions are however immediate to verify for the X n ’s for n ≥ .In §3, we show that every torsion free finite index orientable subgroup of Aut ( F ) can be constructed abstractly by a pinching–and–filling construction, similar to theone given in §1, applied to finitely many tori. It is not clear however how to extendthe explicit procedure given in §1 to describe, e.g., the family of torsion free finiteindex subgroups which are associated with a fixed number of tori.In §4, we explain the origin of the toric presentation given in §1. The presentpaper can be seen as a continuation of an earlier work [3], in which we introducea cobordism category Bord A which can be used to construct groups acting oncomplexes of a given (local) type A . We show below that the techniques of [3] canbe applied to the case of Aut ( F ) . It gives rise groups acting on complexes of type Aut ( F ) as defined in §2.In the case of Aut ( F ) , however, the spaces constructed by surgery in this waymust, by the results in §2, be quotients of the Brady complex X , and the resultingfundamental groups, subgroups of Aut ( F ) . This is not true of many cobordismcategories, and contrasts for example with the categories studied in [3], in which a r X i v : . [ m a t h . G R ] J a n SYLVAIN BARRÉ AND MIKAËL PICHOT the groups accessible by surgery in a given category (of a fixed local type, e.g.,Moebius–Kantor) are typically not pairwise commensurable. Again, the category
Bord A is rather special in this respect when A is the type Aut ( F ) .Finally, we give in §5 an example of a CAT(0) 2-complex X ′ which is locallyisomorphic to the Brady complex to X , but not isometrically isomorphic to it.Here “locally isomorphic” refers to the fact that the links in X ′ are isometric to thelinks in the complex X . Acknowledgement.
The second author is supported by an NSERC discoverygrant. 1.
The toric presentation
Consider the flat torus T of size × defined as follows: A A A A A A A A A A A A A A A A D D B D B D B D B D B B B B B B B B B B B B D D C D C D C D C D C C C C C C C C C C C C D D A D A D A D A D A Every edge in T is oriented and labelled. The boundary is identified in the standardway respecting both the orientation and the labelling of the boundary edges.Note that there is a non trivial Dehn twist, that we will denote τ − , in the verticaldirection.We endow the torus T with the standard Euclidean metric, in which the cellsare (as shown in the figure) lozenges with sides of length 1.Here is the basic construction.The figure contains a total of 20 letters. They are denoted A r , B r , C r , D r , ≤ r ≤ . Let L be a letter. For every triple K of the form K ∶= ( L, L ′ , L ′′ ) consider an oriented triangle with edges labelled by K in the given order. Weattach this triangle to the torus T along its boundary, respecting the orientationand labelling for the boundary edges. This operation, repeated for the twentytriples K , defines a 2-complex B .Let X ∶= ̃ B denote the universal cover of B , and G ∶= π ( B ) denote itsfundamental group of B . URGERY ON
Aut ( F ) Note that the canonical map T → B = X / G is not injective on vertices. Onemay view B as a “wrinkled presentation” of the group G and the map T → B asthe “sewing map”. Observe furthermore that every triple K “jumps” on the torus T . (We call K a “knight”.)By definition, a jump on T is an oriented edge between two vertices of T . Everytriple K defines three jumps, from the extremity of an edge in K to the origin ofthe consecutive edge, modulo 3. Lemma 1.1.
Jumps are either disjoint or they share a common support.Proof.
A jump associated with a triple K corresponds either to the affine transfor-mation ⎧⎪⎪⎨⎪⎪⎩ x ↦ x + y ↦ y − where x is even modulo 6, or to its inverse ⎧⎪⎪⎨⎪⎪⎩ x ↦ x − y ↦ y + where x is odd modulo 6. It is not difficult to show that these two transformationsdo not depend on K . Since they are inverse of each other, jumps with a commonvertex must have the same support. (cid:3) In particular, the jumps define an involution σ of the vertex set of T , whoseorbit partition T /⟨ σ ⟩ coincides with the vertex set of X / G .Let us orient the torus T counterclockwise, and consider the positive labelling ∈ { , , , } of the edges issued from a vertex, where refers to the positive realaxis. The basic construction induces a permutation of the labels associated withevery jump. We shall now describe this permutation. Lemma 1.2.
The permutation of { , , , } associated with the jump ⎧⎪⎪⎨⎪⎪⎩ x ↦ x + y ↦ y − is the 4-cycle ( , , , ) . This shows that the resulting permutation does not depend on K ; the permuta-tion associated with the opposite jump is the inverse permutation. Proof.
Let us for example do the bottom left corner ( , ) , which is mapped to ( , − ) = ( , ) under σ . The corresponding transformation of the counterclockwiselabelling reads ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ = A = A ′ = D ′′ = A ′ ↦ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ = D = A ′′ = A = A ′′ which corresponds to label permutation ( , , , ) . (cid:3) The following shows that X is locally isomorphic isomorphic to the Brady com-plex in the (usual) sense that their links are pairwise isomorphic. Lemma 1.3.
Every link in X is isomorphic to the link of the Brady complex. SYLVAIN BARRÉ AND MIKAËL PICHOT
Proof.
We shall compute the links in X . That it is isomorphic to that of theBrady complex follows from [4, 7] (see also §2 below). Since the expression for σ is independent of the base point in T , it is enough to check the link of the origin.We represent the links at the origin and its image in T as follows (the drawingsrespects the scale provided by the angle metric): The prime labels correspond to the image ( , − ) . According to the previous lemma,edges in the link of X corresponds to the permutation s = ( , , , ) . This definesfour additional edges in the above figure: ( x, s ( x ) ′ ) for every x ∈ { , , , } . It isstraightforward to check that this graph is the link of X (compare §2). (cid:3) The basic construction can be generalized to an arbitrary integer n ≥ in thefollowing way.Suppose first that n is a sufficiently large integer (e.g., n ≥ ). Consider a torus T n of size × n , where the vertical identification involves a Dehn twist τ − . Forevery letter L on (( x, y ) , ( x, y − )) , where x is even, write labels L ′ and L ′′ on,respectively, (( x, y − ) , ( x + , y − )) and (( x + , y − ) , ( x + , y − )) ; for every letter L on (( , y ) , ( , y )) , write labels L ′ and L ′′ on, respectively, (( , y − ) , ( , y − )) and (( , y − ) , ( , y − )) . Then the same construction for every triple K = ( L, L ′ , L ′′ ) on n letters defines a 2-complex B n which is locally isomorphic to the Brady complex.The notation B n is consistent with the previous notation B .One can further extend this construction of B n to every integer n ≥ as follows.Let T ∞ ∶= lim —→ T n (with respect to partial embeddings from a base point) is a cylinderwith an obvious action of Z . Since the set of triples (knights) is Z -invariant, thisaction descends to the basic construction B ∞ ; we let, by definition, X n is theuniversal cover of the quotient B n of this space by n Z . The notation B n is againconsistent. Note however than the description using knights is only clearly visiblefor n suficiently large ( n ≥ is large enough).This shows the following: Proposition 1.4.
The space X n are pairwise isomorphic for n ≥ .Proof. They have a common cover B ∞ . (cid:3) In the next section we give a different proof of this fact, which includes isomor-phism with the Brady complex X .2. Geometric rigidity
We shall describe the local data by a type (or “local type”), following [3, §4]. Inthe latter paper we were interested in two sorts of types, simplicial and metric. Inthe present paper, we shall use labelled types , which add connecting maps to markthe link edges using angle labels as follows (cf. [3, Rem. 4.5]).
URGERY ON
Aut ( F ) Definition 2.1. A labelled type (in dimension 2) is(1) a set of graphs (the links);(2) a set of marked shapes, i.e., polygons with filled interior and labelled angles;(3) a set of connecting maps marking every link edge with an angle label.We define the type Aut ( F ) as follows:(1) the link of the Brady complex; it is isomorphic to the graph (see [7, Fig.6])The letters are associated (see [7, §3] for details) with the presentation ⟨ a, b, c, d, e, f ∣ ba = ae = eb, de = ec = cd,bc = cf = f b, df = f a = ad,ca = ac, ef = f e ⟩ of the braid group B .(2) two shapes, a lozenge and an equilateral triangle, labelled in the followingway: t tt l LL l (3) a connecting map defined by t l t lt l t lL L L L Let T be a labelled type. We say that a 2-complex with labelled face angles is of type T it has the correct links and shapes, and the induced marking of the linkedges corresponds to a connecting map. A homomorphism between two complexesof type T is a 2-complex homomorphism which preserves the angle labels.The following is straightforward to verify from, e.g., the original description of X in [4]. Proposition 2.2.
The Brady complex X is of type Aut ( F ) . Our main theorem in this section is a converse of this statement. More precisely,we prove that the complex X satisfies a universal property : it is freely generatedby any of its faces. SYLVAIN BARRÉ AND MIKAËL PICHOT
Theorem 2.3.
Let X be a 2-complex of type Aut ( F ) . Let S be a face in X andlet f ∶ S → X be a label and shape preserving map from S to a face in X . Thereexists a unique homomorphism ˜ f ∶ X → X whose restriction to S coincides with f .Furthermore, ˜ f is a covering map onto its image. Every 2-complex of type
Aut ( F ) can be naturally endowed with a metric struc-ture, in which the triangle face is equilateral and the lozenge a union of two equi-lateral triangles. By the link condition, every such a complex is locally CAT(0).Every homomorphism between complexes of type Aut ( F ) is isometric, and con-versely, every isometry preserves the angle labels. The universal property in themetric situation states that if f ∶ S → X is an isometry between a face S of X anda face of X , then there exists a unique isometry ˜ f ∶ X → X whose restriction to S coincides with f . Lemma 2.4.
Let X be a 2-complex of type Aut ( F ) . Let S be a face in X and let f ∶ S → X be a map identifying S with a face in X . Let p be a vertex of S . Thereexists a unique label preserving extension ˜ f ∶ St p ( X ) → X of f to the star of p in X .Proof. Let L denote the link of p in X and L the link of f ( p ) in X . The map f .The map f induces a label preserving map from an edge e in L to an edge e in L .Since the labels incident to an arbitrary vertex in L and L are identical, and thelabels around a vertex are pairwise distinct, there exists a unique label preservingextension of f to the faces adjacent to S containing p . More generally, it is easy tocheck that the map e → e admits a unique label preserving extension to a graphisomorphism L → L . This shows that f admits a unique label preserving extension ˜ f ∶ St p ( X ) → X . (cid:3) Proof of Theorem 2.3.
We refer to the standard CAT(0) structure on X definedbefore the lemma. Let C be a maximal ball in X centred in S to which f ad-mits an unique extension. We let f denote this extension. Suppose for towards acontradiction that C has a finite radius.Let p ∈ ∂ C . If p belongs to the interior of a face, it is obvious how to extends f to an ε -neighbourhood of p in X . Suppose that p belongs to the interior of anedge e , and let f be the unique face containing e and intersecting the interior of C . Since both X and X are of type Aut ( F ) , there exists a unique extension of f to an ε -neighbourhood of p in X . Assume now that p is a vertex of ∂ C . In thiscase, C contains a face, and the previous lemma shows that f can be extended ina unique way to an ε -neighbourhood of p .Furthermore, that if p, p ′ are two points in ∂ C at distance ≤ , then the twoextensions of f from p and p ′ coincide on their intersection. Since ∂ C is compact,this shows that f can be extended to an ε -neighbourhood of C , contradicting themaximality of C .Finally, ˜ f is a covering map by construction. (cid:3) Corollary 2.5.
The spaces X n are pairwise isomorphic for every n ≥ .Proof. Since X n is of type Aut ( F ) , we have a covering map X → X n . Since X n is simply connected, this map is an isomorphism. (cid:3) Corollary 2.6.
The groups G n are of finite index in Aut ( F ) . URGERY ON
Aut ( F ) Proof.
The special automorphism group
SAut ( F ) , which is of index 2 in Aut ( F ) ,acts transitively on the set of triangles in X by the description in [7]. If f isa triangle in X , and s an element in G n , then there exists a unique element t s ∈ SAut ( F ) whose restriction to f coincide with s . By the theorem, s and T − s coincide on X , and the map s ↦ t s provides an embedding of G n into SAut ( F ) . (cid:3) Another corollary, Theorem 2.8 below, shows that the Brady complex admits a“frame” in the following sense.We recall that a flat plane in X is an isometric embedding R ↪ X of thestandard Euclidean plane in X . Definition 2.7. A frame on X is an orientation, and a labelling by two letters e and f , of the edge set of X , such that for every flat plane Π in X which is aunion of lozenges, the following holds(1) the ordered set B x ∶= ( e x , f x ) of outgoing edges at a vertex x in Π , withrespective labels e and f , forms a basis of Π ,(2) the unique translation of Π which takes a vertex x to a vertex y takes theordered set B x to the ordered set B y .Thus a frame is a way to move a basis consistently along the various embeddings Π into X . Theorem 2.8.
There exists a frame on X .Proof. The map X ↠ B is a covering map. We consider the obvious frame on thetorus T , and the induced orientation and labelling of the edge set of B by thesewing map T → B (which is a bijection on the edge set), and lift the orientationand labelling to X using the map X ↠ B . Since every flat plane in X mapsonto the image of T in B , this defines a frame on X . (cid:3) We shall say that a group of automorphisms of X is orientable if it preservesthe frame constructed in Theorem 2.8.Note that every finite index subgroup of Aut ( F ) contains a finite index subgroupwhich is orientable. Proof.
Let G be a finite index subgroup of Aut ( F ) . Then the group G ∩ G is offinite index in Aut ( F ) : [ Aut ( F ) ∶ G ∩ G ] ≤ [ Aut ( F ) ∶ G ][ Aut ( F ) ∶ G ] . Furthermore, G ∩ G is orientable since G is. (cid:3) Pinching and filling tori
The spaces in §1 are obtained in two steps, by a procedure which can be describedas “pinching and systolic filling” starting from a flat torus.Theorem 2.3 shows that every such a construction, using a family of flat tori, willhave X as a universal cover, provided it satisfies a few basic conditions, describedin the following proposition. Proposition 3.1.
Let t ≥ be an integer. Suppose that:(1) T , . . . T t is a finite family of flat tori, endowed with a simplicial metricstructure in which every cell is a lozenge with sides of length 1 SYLVAIN BARRÉ AND MIKAËL PICHOT (2) σ is a fixed point free involution on the vertex set of T ∶= ⊔ tk = T k (3) the systolic length in T ′ ∶= T /⟨ σ ⟩ is 3(4) every edge in T ′ belongs to a unique systole of length 3(5) the systolic filling B of T ′ , obtained by attaching isometrically an equilateraltriangle to every systole in T ′ , is locally CAT(0) (i.e., the link girth in B is ≥ π )then ̃ B ≃ X .Proof. By Theorem 2.3 it is enough to prove that B is of type Aut ( F ) . Since σ isfixed point free, the link at a vertex in B contains a union of two disjoint circles oflength π .We shall use the notation in Lemma 1.3. Since every edge belongs to a uniquesystole of length 3, the systolic filling provides an involution τ of the set { , , , }⊔{ ′ , ′ , ′ , ′ } of vertices in the link. Since B is locally CAT(0), and the edge lengthfrom the systoles are π / , the involution τ induces a bijection from { , , , } to { ′ , ′ , ′ , ′ } . We may assume without loss of generality that τ ( ) = ′ . By theCAT(0) condition, it follows that τ ( ) ≠ ′ .Suppose that τ ( ) = ′ . Then the distance between and ′ is ≤ π , which implies τ ( ) = ′ and τ ( ) = ′ . In this case, however, the cycle ′ ′ is of length < π ,which is a contradiction. Thus, τ ( ) ≠ ′ . This implies that τ ( ) = ′ . Applyingagain the CAT(0) condition, we must have τ ( ) = ′ and τ ( ) = ′ .Labelling the angles of the faces as in §2, the above shows that the link of vertexin B is label isomorphic to the link of type Aut ( F ) , where the labels t are associatedwith the systolic filling. This implies that B is of type Aut ( F ) and therefore that X ≃ ˜ B . (cid:3) Furthermore, every (sufficiently deep) orientable torsion free subgroup of finiteindex is constructed in this way:
Proposition 3.2.
Let G be an orientable torsion free subgroup of finite index in Aut ( F ) . Suppose that the injectivity radius of X / G is > . Then there existsa finite family of tori T , . . . , T t and a fixed point free involution σ on the vertexset of T ∶= ⊔ tk = T k , such that T ′ ∶= T /⟨ σ ⟩ satisfies the condition in the previousproposition, and the systolic filling B of T ′ is isometric to X / G .Proof. We say that two edges e and f in X (or X / G ) are equivalent if there existsa gallery ( f , . . . , f n ) containing them, such that f i is a losenge for every i .Let e be an edge in X and ˜ e be a lift of e in X . It is clear that the equivalenceclass [ ˜ e ] of ˜ e maps surjectively onto the equivalence class of [ e ] under the coveringmap π ∶ X ↠ X / G . The convex hull H of [ ˜ e ] is isometric to a flat plane tessellatedby lozenges. We let T ′ e denote the image of H under π .Say that a vertex x ∈ T ′ e is a double point if the link of T ′ e at x is a disjoint unionof circles. The map H → T ′ e factorize through a map H → T e → T ′ e , where T e isobtained from T ′ e by blowing up every double point. Since X / G is compact, so is T ′ e . Therefore, T e is compact. Since H ↠ T e is a covering map and G is orientable,it follows that T e is a torus. We let σ e be the partially defined involution on T e inducing the quotient map T e ↠ T ′ e .Let e , . . . , e t be a representative set of equivalence classes of edges in X / G .Associated with the e k ’s are tori T k and partially defined involution σ k on T k suchthat the edge set of T k /⟨ σ ⟩ k ⊂ X / G coincides with the equivalence class of e k . URGERY ON
Aut ( F ) Furthermore, for every vertex x ∈ T k not in the domain of σ k , there exists aunique k ′ ≠ k such that x is a vertex of T ′ k . This defines an involution σ on thecomplement of ⊔ k σ to itself on T ∶= ⊔ k T k . This involution is a fixed point freeinvolution on T . Since the injectivity radius of X / G is > , the systolic length of T /⟨ σ ⟩ is ≥ , and therefore X / G is the systolic completion of T /⟨ σ ⟩ in the sense ofthe previous proposition. (cid:3) The map T → B can be viewed as a structure of “space with jumps” on thetorus (or union of tori) T . The geodesics with respect to such a structure in T areallowed to jump between certain transverse codimension 1 subspaces they cross (inthe present situation, it is the 1-skeleton, which are the sides of the triangles). Thelength of the jump, and the incidence angles are described by the geometry of theadded triangles. A pinching occurs along a codimension 2 subspaces (intersection ofcodimension 1 subspaces), which are singular sets, corresponding to instantaneousjumps of a geodesic between two points in T . We will not attempt to formalize thisnotion further in the present paper. Remark 3.3.
The number t of tori, and the geometric parameters of the individualtori, provide conjugacy invariants for the given subgroup G . As mentioned in theintroduction, the description of the family of subgroups with a prescribed invariant,e.g., the torsion free finite index subgroups G of Aut ( F ) with a given torus number t ( G ) , seems rather involved however.4. A group cobordism for
Aut ( F ) In this section we show that the surgery techniques from [3] which were used toconstruct (in many cases, infinitely many) groups of a given type, can be appliedto the group
Aut ( F ) . (Indeed, this is how the toric presentation in §1 and thegroups G n were found.)Let A be a (e.g., labelled) type. A category Bord A of group cobordisms of type A can be defined as follows. The objects in this category are called collars, and thearrows, group cobordisms; in the present paper we only discuss the case where A is the type Aut ( F ) defined in §2.Let us first review the notion of collar. An (abstract) open collar is a topologicalspace of the form H ×( , ) where H is a graph (not necessarily connected). If X is a2-complex, an open collar in X is, by definition, an embedding C ∶ H ×( , ) ↪ X . Weshall refer to the domain H ×( , ) as the abstract collar defining C . The dual of anopen collar of X is the open collar C ′ ∶ H ×( , ) ↪ X defined by C ′ ( x, t ) ∶= C ( x, − t ) .The collar closure of C the topological closure C of the image of C in X ; the span of C in X is the set span ( C ) of vertices of X contained in collar closure of C ;the simplicial closure of C is is the union of all the open edges and open faces itintersects. As in [3] we only consider collars which are simplicially closed and vertexfree.We shall denote by Bord
Aut ( F ) the category of group cobordisms of type Aut ( F ) .We construct an object C in Bord
Aut ( F ) as follows.Fix an integer y ∈ N . We use the notation introduced at the end of §1. Wewill view C as a “slice” of the cylinder T ∞ . We fix four letters A y , B y , C y , D y respectively on (( x, y ) , ( x, y − )) where x = , , and on (( , y ) , ( , y )) . Recallthat for every letter L on (( x, y ) , ( x, y − )) , where x is even, we write labels L ′ and L ′′ on, respectively, (( x, y − ) , ( x + , y − )) and (( x + , y − ) , ( x + , y − )) , while for a letter L on (( , y ) , ( , y )) , we write labels L ′ and L ′′ on, respectively, (( , y − ) , ( , y − )) and (( , y − ) , ( , y − )) .By definition, the cylinder T ∞ is a quotient of a strip [ , ] × R using the twist τ − in the vertical direction. Recall that a gallery is a sequence of faces ( f , . . . , f n ) such that f i ∩ f i + is an edge.We say that a gallery in T ∞ is generating if it is closed (i.e., cyclic permutationsremain galleries) and homotopic to an element generating π ( T ∞ ) . Lemma 4.1.
The minimal generating gallery has length n = .Proof. Indeed, writing T ∞ as a quotient of a strip [ , ] × R of size × ∞ by τ − ,the gallery distance between a boundary edge on { } × R and its image by τ − in { } × R is = + . (cid:3) The collar C will be built from a minimal generating gallery on T ∞ . Startingfrom the edge labelled A y , the gallery is defined by the succession of edges f i ∩ f i + .The edges have the following labels: A y , A ′ y + , A ′ y , A ′′ y + , B y − , B ′ y − , B ′ y − , B ′′ y − , C y − , C ′ y − , C ′ y − , C ′′ y − Note the corresponding gallery ( f , . . . , f ) is closed: every change of letter occurswith a drop of − for a total drop of − , which is consistent with τ − . This definesa “zig-zag” gallery generating π ( T ∞ ) .As a topological space the gallery ( f , . . . , f ) is homeomorphic to [ , ] × S .We shall refer to the gallery minus its boundary as open. Definition 4.2.
Let C be the union of(1) the image of the open generating gallery ( f , . . . , f ) in the basic construc-tion B ∞ .(2) the triangles in B ∞ associated with the following six triples (knights) K =( L, L ′ , L ′′ ) on the letters L = A y + , A y , B y − , B y − , C y − , C y − where every triangle associated with a triple K is semi-open, in the sensethat it does not contain the (unique) edge not belonging to the image ofthe gallery. Lemma 4.3. C is a product space.Proof. It is clear that the open gallery is a product space homeomorphic to ( , ) × S . Under this identification, the added triples K define a space of the form ( , )× H where H is a finite graph (the nerve) obtained by adding 6 edges to S . (cid:3) One can of course give an explicit description of H : Lemma 4.4.
The graph H is isomorphic to the Cayley graph of Z / Z , with respectto 1, together with an additional edge ( n, n + ) for every n ≡ , . Therefore, we may view C as a open collar in B ∞ under the identity mapping C → B ∞ . Lemma 4.5. C is a full collar in B ∞ Proof.
Every open edge e = f i ∩ f i + in C belongs to a (unique) triple K , andtherefore every point in e has an open neighbourhood included in C . (cid:3) URGERY ON
Aut ( F ) Since B ∞ is a complex of type Aut ( F ) , the above shows that the isomorphismclass of C is an object in the category Bord
Aut ( F ) .The arrows in Bord
Aut ( F ) are group cobordisms: Definition 4.6. A group cobordism is a 2-complex B together with a pair ( C, D ) of collars of B whose boundaries ∂ − C and ∂ + D form a partition of the topologicalboundary of B : ∂ B = ∂ − C ⊔ ∂ + D. Let us construct the group cobordism B of type Aut ( F ) . The collar C dependson y ∈ N , however, it is clear that C y ≃ C y + . The cobordism B has C as domainand codomain. Definition 4.7.
Let B be the union of(1) C y ∪ C y + (2) the closed triangle in B ∞ associated with the triple K = ( L, L ′ , L ′′ ) on theletters L = D y − .Again, B depends on y , where B y is isomorphic to B y + and defines a uniquearrow, again denoted B , in Bord
Aut ( F ) . The inclusion map L B , R B ∶ C → B (leftand right collar boundary) and the obvious inclusion of C as C y and C y + .In particular: Theorem 4.8.
The map taking 1 to B induces a unital inclusion N → Bord
Aut ( F ) .Proof. Indeed, B ○ n ≠ B ○ m if n ≠ m , where B ○ n refers to the n -fold composition B ○ ⋯ ○ B in Bord
Aut ( F ) . (cid:3) In the language of [3], the above shows the following:
Theorem 4.9.
Aut ( F ) is virtually accessible by surgery. This means that
Aut ( F ) admits a finite index subgroup which is the fundamen-tal group of a complex obtained by a surgery construction in a cobordism category(see [3, §10]). Here the groups G n are of finite index in Aut ( F ) and the fundamen-tal groups of the complexes B n , which are of type Aut ( F ) defined by a surgeryconstruction in Bord
Aut ( F ) .We take this opportunity to make a correction to [2, Lemma 17]. At the bottomof the page it is stated that “there are two extensions of this section”: it should be“three extensions”. Namely, in the first case (when the lozenges on the south-easttriangles are oriented pointing south) one extension is the 3-strip, as indicated,which amounts to extending the lozenges with two triangles. A third sort of ex-tension uses lozenges instead. In this case, the lozenges belong to a (using theterminology in [2]) semi-infinite ◇ -strip of type × ∞ . This can be visualized us-ing the surgery construction above: starting from the closed triangle defined in B above, Def. 4.7, (2), one may use three lozenges belonging to a single collar (eitherall belonging to C y , or all in C y + ) which can be extended into three semi-infinite ◇ -strip of type × ∞ in the universal cover (so the resulting puzzle has an order 3symmetry). 5. Complements to Theorem 2.3
We conclude some remarks on Theorem 2.3, regarding spaces locally isometricto X . It is an interesting exercise to construct groups acting freely uniformly on a CAT(0) 2-complex locally isometric to the 2-complex X of Aut ( F ) (but notisometric to it), in the sense that their link are isometric to the link of X . In thepresent section, we provide one example.By Theorem 2.3 such a complex X ′ is not of type Aut ( F ) . The example willbe of the following type.Let A denote the metric type (i.e., a set of metric graphs, and a sets of shapes)defined by:(1) Graph: the link of the Brady complex with the angular metric (see §2).(2) Shapes: an equilateral triangle, and an hexagon with sides of length 1. Bothare viewed as standard polygons in the Euclidean plane with the inducedmetric.By definition, every CAT(0) 2-complex of type A is locally isometric to X butnot isometric to it. Proposition 5.1.
There exists a group G ′ acting freely uniformly isometrically ona CAT(0) 2-complex X ′ of type A . The construction is as follows. We begin with a single hexagon on a set of 6vertices, which we denote { + , − , + , − , + , − } , and edges labelled from 1 to 6 in a cyclic order as follows. + − + − + − We shall realize these 6 vertices as the vertex set of a locally CAT(0) space oftype A , containing the hexagon as a face.Consider additional edges between these vertices:two edges between i − and i + two edges between i + and ( i + ) + two edges between i − and ( i + ) − (where i is an index modulo 3) organized and named as follows: URGERY ON
Aut ( F ) + + + + − − − − a b c d a b c d c d a b e f e f e f Together with the edges of the hexagon, this defines a regular graph of order 8.Note that this graph has a natural symmetry σ of order 3 taking evert letter l i tothe letter l i + (modulo 3).Consider the following hexagon and four triangles ( a , c , f , e , d , b )( d , a , ) ( f , d , − )( b , c , − ) ( c , e , ) Together with their images under σ , this defines 3 triangles and 12 triangles. Inaddition to these triangle add the four triangles: ( a , a , a ) ( b , b , b )( e , e , e ) ( f , f , f ) This defines a 2-complex, whose fundamental group is G ′ and universal cover X ′ .It is immediate to check that: Lemma 5.2.
The link of X ′ is isometric to the link of X .Proof. Note that it is enough to check a single vertex, since σ and the reflectionwith respect to the horizontal axis extend to the 2-complex.We may index the vertex set of the link by a , b a , b , c , d , , , where the lattertwo numbers are associated with the initial hexagon. There are four hexagon edges: ( a , b ) , ( a , c ) , ( b , d ) , and ( , ) (for the first hexagon). One can then draw theedge associated with triangles: these are ( a , d ) , ( d , ) , ( b , c ) , ( c , ) , from theimages under σ : ( b , ) , ( a , ) , and finally, ( a , a ) and ( b , b ) .It is not difficult to show that this graph is isometric to the link of X . (cid:3) We also note that:
Proposition 5.3.
Aut ( X ′ ) is vertex transitive. This is part of the argument in the previous lemma.
References [1] Barré, S., Pichot, M., Intermediate rank and property RD, arXiv:0710.1514.[2] Barré, S., Pichot, M.,
Aut ( F ) puzzles. Geometriae Dedicata, 199(1), 225-246, (2019).[3] Barré, S., Pichot, M., Surgery on discrete groups. Preprint.[4] T. Brady. Automatic structures on Aut ( F ) . Archiv der Mathematik , 63(2):97–102, 1994.[5] T. Brady. Artin groups of finite type with three generators.
The Michigan MathematicalJournal , 47(2):313–324, 2000.[6] Bridson, M.R. and Haefliger, A., Metric spaces of non-positive curvature (Vol. 319). SpringerScience & Business Media (2013).[7] J. Crisp, L. Paoluzzi. On the classification of CAT(0) structures for the 4-string braid group.
The Michigan Mathematical Journal , 53(1):133–163, 2005.
Sylvain Barré, UMR 6205, LMBA, Université de Bretagne-Sud,BP 573, 56017,Vannes, France
Email address : [email protected]
Mikaël Pichot, McGill University, 805 Sherbrooke St W., Montréal, QC H3A0B9, Canada
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