Automorphism groups of Coxeter groups do not have Kazhdan's property (T)
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Varghese, Olga
We show that for a large class w of Coxeter groups the following holds : given a group WΓ in W, the automorphism group Aut(WΓ) does not satisfy Kazhdan's property (T).
 Before I start, I want to thank Indira, Lewis and Karen for organizing this conference. So today I want to talk about cocceter groups and Kashtan's property. So in particular, I want to share with you some results regarding the conjecture that automorphism groups of cocceter groups do not have property T. So as the main objects in my talk are cocceter groups, I start with the definition of this object. So given a finite, simplification graph gamma with an H labeling, the cocceter group W gamma associated with gamma is a group with the presentation, with the following presentation. So the generator set is equal to the vertex set. For each vertex V, we have a relation V square. And for each H in the graph, we have a relation VW to the power of the H label. So let us spend some time at examples. So this graph has two vertices and one H. So the corresponding cocceter group has two generators and three relations. And one can show that the algebraic structure of this group is as follows. So the cocceter group is a direct product of two copies of the two. And we can also describe this group as an isometry group of a geometric object, so of a two colored square. So we only allow reflections. So one more example. Again we have a graph with two vertices and one H. So the corresponding cocceter group has again two generators and three relations. So the algebraic structure of this group is as follows. So one can show that this group is a semi direct product of Z5 with the two. And this group can also be described as the isometry group of a geometric object of the pentagon. Okay, so one more example. So this graph has two edges. Hence the corresponding cocceter group is equal to the free product of the two. In this graph no edge is missing and all edge labels are equal to two. Hence the corresponding cocceter group is the direct product of the two. So if all edge labels of the graph are equal to, then the cocceter group is called right angled. Also you can construct a cube complex on that the group acts nicely. So usually one says that right angle cocceter groups interpolate between free product of the two and direct product of the two. Okay one more example. So the corresponding cocceter group of this graph has a following structure. So for experts this cocceter group is the affine cocceter group of type A3 children. It can be geometrically described as the isometry group of a very nice styling of R3. So let's talk about properties of cocceter groups. So cocceter groups were named after Harold Scott Mcdonald cocceter. So it is obvious that cocceter groups are finally presented. Finite cocceter groups were classified in terms of Dinkin diagrams by cocceter. The cocceter groups are linear groups, cocceter groups are zero groups and a cocceter group W gamma is chroma for parabolic if and only if the poison non-hapabolic group Z times Z is not a subgroup of W gamma. The next fact a cocceter group is meanable if and only if the free group of two generators is not a subgroup of W gamma. Okay and the next fact so a cocceter group is either virtually free abelian or large. So today our focus is on the fixed point properties. So in particular on Seas property of A and Castan property T. So let us first recall the definition of property of A. So let A be the class of simplisher trees. A group G is said to have Seas property of A if any simplisher action of G without inversion on any member of A has a fixed point. So it follows from Seas results that a cocceter group W gamma has Seas property of A if and only if gamma is a complete graph. So no H in the defining graph is missing. Next definition. So let H be the class of free wheeled wheeled spaces. A countable group G is said to have Castan's property T if any action of G by azometries on any member of H has a fixed point. So property T was introduced by Castan for topological groups in terms of unitary representations. So we are not going to recall the original definition. The definition of property T in terms of fixed points was given by the law and key have that for the last fact about cocceter groups is as follows. The cocceter group W gamma has Castan's property if and only if the corresponding cocceter group so W gamma is finite. So the proof by but cycle Janusz Kiewicz and Spatia has a geometric flavor. So they used the action of a cocceter group on the corresponding cocceter complex. So they constructed a really nice embedding of the cocceter complex in a Hilbert space in such a way that the cocceter group acts by azometries with respect to the induced metric. So many properties of a cocceter group can be checked easily on the graph gamma. So let us figure out some properties of the right angle cocceter group W gamma where the graph is a Peterson graph. Okay, so assume that all H labels are two, then the corresponding right angle cocceter group is infinite. So that's easy. So this follows from the fact that a right angle cocceter group is finite if and only if the defining graph is complete. The group is hyperbolic. So it was proven by Musong that a right angle cocceter group is hyperbolic if and only if gamma has no squares as full subgraphs. And it is obvious that the Peterson graph has no squares. Okay, so cocceter group is not virtually free. So virtually free cocceter groups were described by Michalek and Chanst. So they showed that a right angle cocceter group is virtually free if and only if it has no cycles of lengths greater or equal four. So the Peterson graph has a pentagon as a subgraph here. Hence, the corresponding cocceter group is not virtually free. Okay, it has no t because the group is infinite. And the Peterson graph is not complete, hence no fA. But the automorphous group of W gamma has fA. So this result was recently proven by Neuse-Leder in his PhD thesis. Okay, so our focus today is on the fixed-band property t. So let us proceed with 10 facts about t. So property t was introduced by Kashtang for locally compact groups in terms of unitary representations. So what can property t tell us about the group? So Kashtang proved that if a locally compact group has t, then it is compactly generated. And if a group has t, then all its lattices satisfy t. Hence all lattices of a t group are finally generated. Okay, so Kashtang's motivation was to use this property to prove that a large class of lattices in lea groups are finally generated. Okay, so the lea group SLNR has t, hence all lattices in SLNR are finally generated. So for example, it is not hard to prove that all lattices in the group consisting of real numbers are finally generated, but it is not so easy to show that all lattices in nice lea groups are finally generated. Okay, so SLNZ is the lattice in SLNR, hence has t. Okay, so Kashtang's property t is inherited by quotients and is preserved undertaking finite index subgroups. If a group G acts fixed on freely on a tree, then G does not satisfy property t. So this connection between property t and CS property of A was proven independently by Alperin and Vatatani. So trees are one-dimensional KDL cube complexes. And nebulant trees generalize this factor higher-dimensional cube complexes. So they proved if a group G acts fixed on freely on a KDL cube complex, then G does not satisfy property t. And this fact was recently used by Osaida to show that burnt-side groups do not have t. So we are mostly interested in discrete groups of property t and we have the following motivation to search for a discrete t group. So if a nice discrete group G has t, then one can construct a family of really nice graphs, so a family of expander using calligraphs of finite quotients. So I like groups and graphs and other geometric objects and I like to construct nice geometric objects using groups. And this fact is enough motivation for me to search for t groups. So free groups and free Abellion groups do not have property t, but the automaphy's group offer free group has t and the automaphy's group offer free Abellion group has t. So coxsider groups are well understood objects in geometric group theory. So they provide interesting examples and counter examples to many conjectures. And I think it is natural to ask if the automaphy's group of a nice coxsider group has t. So we conjecture that this is not the case. So the automaphy's group of an infinite coxsider group does not have Kashtan's property t. So some results in this direction will let w gamma be an infinite coxsider group. So if gamma is discrete, so if gamma has no edges, then out w gamma does not have property t. And this result holds for all right angled coxsider groups. So if w gamma is a right angled one, then the automaphy's group does not have property t. Okay, so let's talk about methods to prove that a group does not have t. So if the outer automaphy's group is finite, then the automaphy's group of the coxsider group does not have t. So why? So this follows from the fact that the center of a coxsider group is always finite and the quotient w gamma, center of w gamma does not have t. So it was proven by Howlett, Rowley and Taylor that if gamma is complete, then the outer automaphy's group of the coxsider group is finite, hence the automaphy's group has no t. Okay, so Nibbler and Reeves constructed for each coxsider group a kazero cube complex and which the group acts fixed on freely. So what can we do with this complex? So we can try to extend the action of the coxsider group w gamma on the corresponding Nibbler-Reeves-Kazero-Cube complex to an action of a finite index subgroup of out w gamma. So for some coxsider groups, this is doable, but for a general coxsider group, we don't even know if the automaphy's group is finitely generated or not. Okay, so we can try to show that the automaphy's group maps virtually onto w gamma or onto a special infinite subgroup of w gamma, and this is what we are going to do. So let me state the main theorem. So let w gamma be an infinite coxsider group. If gamma has a maximal complete subgraph delta such that the center of w delta is trivial, so this assumption can be checked easily on the graph gamma. So then the automaphy's group virtually subjects onto w gamma. And in that case, the automaphy's group does not have t. Okay, the second result is as follows. So let w gamma be an infinite coxsider group. If there exist two non-adjacent even vertices, vw and gamma, then the automaphy's group virtually subjects onto a full product of the two. So in particular, no t for the automaphy's group. And what do I mean by an even vertex? So a vertex, v in v, is called even if all h labels of e in e with v in e are even. So let us discuss some examples. Okay, so the corresponding coxsider group of this graph is the right angled one and it is infinite. So hence the group has no t. So what about this graph? Okay, so this graph has two non-adjacent even vertices. So this vertex is an even one and this vertex is an even one. Hence the group out w gamma has no t by part two of the main theorem. Next example, so this graph has only one even vertex, this one. So we cannot apply part two of the main theorem, but this graph has a maximum complete subgraph data such that w data is an irreducible infinite coxsider group. So let us consider this triangle, so triangle is the maximum complete subgraph in the graph gamma. The corresponding special coxsider subgroup is an alpha and one of type A2 tilde. It is irreducible and irreducible coxsider groups have all this trivial center. Hence the automorphism has no t by part one of the main theorem. Okay, so more results. So let w gamma be an infinite coxsider group again. So if gamma has no cycle of length three, then the automorphism has no t. So why? Okay, so in this case, maximum complete subgraphs in gamma are edges. And if gamma has an edge where the edge label is odd, then the center of this corresponding special subgroup is trivial and we can apply part one of the main theorem to see that the automorphism group has no t. So otherwise, all edge labels are even and by part two of the main theorem follows that the automorphism group has no t. Okay, so if all edge labels are even, then the automorphism group does not have property t. So why? So if gamma is complete, then the outer automorphism group is finite. Hence the automorphism group has no t. And if gamma is not complete, then gamma has two non-adjacent even vertices, hence by part two of the main theorem, the automorphism group does not satisfy property t. So if all edge labels are at least three, then out w gamma does not have property t. So it follows from a characterization of finite coxseta groups that if gamma has at least three vertices and all edge labels are at least three, then all special subgroups of w gamma are infinite and irreducible. So all non-trivial special subgroups, so all infinite special subgroups of w gamma are irreducible. So irreducible infinite coxseta groups have all the three real center. So in particular, gamma has a maximal complete subgraph such that the corresponding special subgroup has three real center. Hence by part one of the main theorem, the automorphism group has no t. Okay, so one more example. So what about this graph? Okay, so this graph has two connected components and has no even vertices. So maximal complete subgraphs are triangles. So we have eight triangles and the corresponding special subgroup of a triangle has non-trivial center. So we cannot apply the results in the main theorem to this automorphism group. So question, does out w gamma satisfy property t? We don't know. Okay, so back to the main theorem. And today I want to discuss the main steps in the proof of part one. Okay, so an important puzzle piece of the proof is the following lemma. So let gamma be a coxseta graph and delta one to delta n, the maximal complete subgraph. We define w delta i in brackets as the following set. Then the following map is a well-defined homomorphism. And in particular the kernel of phi has finite index in the automorphism group. Okay, so the idea of the proof is based on the following fact. Maximal FA subgroups of a coxseta group w gamma are exactly conjugates of special subgroups w delta i. And this result was proved by Mihalyk and Chanst. So for w delta i, the group, so the image of w delta i under F is again a maximal FA subgroup. So we know that the existing J such that we have this relation. So such that as image of w delta i under F is in this set. And then one can prove that the image of F is indeed a projection and phi is a homomorphism. Okay, one remark. So for F in the kernel of phi, the existing A in w gamma such that the image of w delta i under F is a conjugate of w delta i by A. And we can compose the map F with C A inverse where C A inverse is a conjugation by A inverse. And then C A inverse composed with F is an automorphism of w delta i. Okay, so back to the theorem and let us discuss the main steps in the proof. Okay, first we construct a map psi from kernel of phi to the outer automorphism group of w delta. And then we show that the kernel of psi maps onto w gamma. So the kernel of psi has finite index in the auto, the automorphism group of w gamma and w gamma does not satisfy property t. Since the group out w gamma does not have property t neither. Okay, so let us discuss the main step is a proof more precisely. So we already know that for F in the kernel of psi, the existing A in w gamma such that the image of w delta under F is a conjugate of w delta by A. And then we set the image of F under psi equal C A inverse composed with F and we take the conjugacy class where C A inverse is again conjugation by A inverse. And this map is a well-defined homomorphism, we'll have to check it. So then we construct a map psi from kernel of psi to w gamma. So for F in the kernel of psi, the existing B in w gamma such that the image of w under F is a conjugation by B or all w in w delta and we set psi F equal B. Okay, so this map is a well-defined one. So why? Because this element B is unique with this property. Okay, so assume that the existing C in w gamma such that the image of w under F is a conjugation by C. And then B inverse C is in the centralizer of w delta in w gamma. And this centralizer is equal to the center of w delta and the center of w delta is by a sumptuous trivial. So B is equal C and one can also prove that the map psi is an epimorphism. Okay. Okay, so recall a group G is called large if it virtually subjects onto the free group of two generators. So it is known that any coccister group is either virtually abelian or large. So moreover, these properties can be verified easily on the coccister graph. So concerning largeness of automorphism groups of coccister groups, so it was proven by Zalle and Zuse that the auto-automorphism group of a right angled coccister group is either large or virtually abelian. So we obtain the following result concerning largeness of automorphism groups. So let's w gamma be a large coccister group. If gamma has a maximum complete subgraph delta such that the center of w delta is trivial, then the group out w gamma is large. Really nice result. So what's next? A very interesting class of groups is the class consisting of right angled arting groups. So let us give a quick definition. So given a finite simplisher of graph gamma, the right angled arting group A gamma is defined as follows. So the generator set is equal to the vertex set of the graph and for each edge in the graph we have one relation. So for each edge, uv, we have a relation uv, u minus 1, v to the power of minus 1. So let us proceed with some examples regarding property T of automorphism group of a right angled arting group. So the group out of 5 has T. For n bigger or equal to 6, the group out of n has T, but the group out AC5 where C5 is the pentagon does not have T. And there are more examples in the literature. I plan to answer the following question. So for which shape of the graph gamma does the automorphism group out A gamma have Kashtan's property T? So I have results for small graphs, but these results will be the topic for my next talk. So thank you for your attention.
 Before I start, I want to thank Indira, Lewis and Karen for organizing this conference.  So today I want to talk about cocceter groups and Kashtan's property.  So in particular, I want to share with you some results regarding the conjecture that  automorphism groups of cocceter groups do not have property T.  So as the main objects in my talk are cocceter groups, I start with the definition of this  object.  So given a finite, simplification graph gamma with an H labeling, the cocceter group W gamma  associated with gamma is a group with the presentation, with the following presentation.  So the generator set is equal to the vertex set.  For each vertex V, we have a relation V square.  And for each H in the graph, we have a relation VW to the power of the H label.  So let us spend some time at examples.  So this graph has two vertices and one H. So the corresponding cocceter group has two generators  and three relations.  And one can show that the algebraic structure of this group is as follows.  So the cocceter group is a direct product of two copies of the two.  And we can also describe this group as an isometry group of a geometric object, so of  a two colored square.  So we only allow reflections.  So one more example.  Again we have a graph with two vertices and one H. So the corresponding cocceter group  has again two generators and three relations.  So the algebraic structure of this group is as follows.  So one can show that this group is a semi direct product of Z5 with the two.  And this group can also be described as the isometry group of a geometric object of the  pentagon.  Okay, so one more example.  So this graph has two edges.  Hence the corresponding cocceter group is equal to the free product of the two.  In this graph no edge is missing and all edge labels are equal to two.  Hence the corresponding cocceter group is the direct product of the two.  So if all edge labels of the graph are equal to, then the cocceter group is called right  angled.  Also you can construct a cube complex on that the group acts nicely.  So usually one says that right angle cocceter groups interpolate between free product of  the two and direct product of the two.  Okay one more example.  So the corresponding cocceter group of this graph has a following structure.  So for experts this cocceter group is the affine cocceter group of type A3 children.  It can be geometrically described as the isometry group of a very nice styling of R3.  So let's talk about properties of cocceter groups.  So cocceter groups were named after Harold Scott Mcdonald cocceter.  So it is obvious that cocceter groups are finally presented.  Finite cocceter groups were classified in terms of Dinkin diagrams by cocceter.  The cocceter groups are linear groups, cocceter groups are zero groups and a cocceter group  W gamma is chroma for parabolic if and only if the poison non-hapabolic group Z times  Z is not a subgroup of W gamma.  The next fact a cocceter group is meanable if and only if the free group of two generators  is not a subgroup of W gamma.  Okay and the next fact so a cocceter group is either virtually free abelian or large.  So today our focus is on the fixed point properties.  So in particular on Seas property of A and Castan property T.  So let us first recall the definition of property of A.  So let A be the class of simplisher trees.  A group G is said to have Seas property of A if any simplisher action of G without inversion  on any member of A has a fixed point.  So it follows from Seas results that a cocceter group W gamma has Seas property of A if and  only if gamma is a complete graph.  So no H in the defining graph is missing.  Next definition.  So let H be the class of free wheeled wheeled spaces.  A countable group G is said to have Castan's property T if any action of G by azometries  on any member of H has a fixed point.  So property T was introduced by Castan for topological groups in terms of unitary representations.  So we are not going to recall the original definition.  The definition of property T in terms of fixed points was given by the law and key have that  for the last fact about cocceter groups is as follows.  The cocceter group W gamma has Castan's property if and only if the corresponding cocceter  group so W gamma is finite.  So the proof by but cycle Janusz Kiewicz and Spatia has a geometric flavor.  So they used the action of a cocceter group on the corresponding cocceter complex.  So they constructed a really nice embedding of the cocceter complex in a Hilbert space  in such a way that the cocceter group acts by azometries with respect to the induced  metric.  So many properties of a cocceter group can be checked easily on the graph gamma.  So let us figure out some properties of the right angle cocceter group W gamma where the  graph is a Peterson graph.  Okay, so assume that all H labels are two, then the corresponding right angle cocceter  group is infinite.  So that's easy.  So this follows from the fact that a right angle cocceter group is finite if and only  if the defining graph is complete.  The group is hyperbolic.  So it was proven by Musong that a right angle cocceter group is hyperbolic if and only if  gamma has no squares as full subgraphs.  And it is obvious that the Peterson graph has no squares.  Okay, so cocceter group is not virtually free.  So virtually free cocceter groups were described by Michalek and Chanst.  So they showed that a right angle cocceter group is virtually free if and only if it has  no cycles of lengths greater or equal four.  So the Peterson graph has a pentagon as a subgraph here.  Hence, the corresponding cocceter group is not virtually free.  Okay, it has no t because the group is infinite.  And the Peterson graph is not complete, hence no fA.  But the automorphous group of W gamma has fA.  So this result was recently proven by Neuse-Leder in his PhD thesis.  Okay, so our focus today is on the fixed-band property t.  So let us proceed with 10 facts about t.  So property t was introduced by Kashtang for locally compact groups in terms of unitary  representations.  So what can property t tell us about the group?  So Kashtang proved that if a locally compact group has t, then it is compactly generated.  And if a group has t, then all its lattices satisfy t.  Hence all lattices of a t group are finally generated.  Okay, so Kashtang's motivation was to use this property to prove that a large class  of lattices in lea groups are finally generated.  Okay, so the lea group SLNR has t, hence all lattices in SLNR are finally generated.  So for example, it is not hard to prove that all lattices in the group consisting of real  numbers are finally generated, but it is not so easy to show that all lattices in nice  lea groups are finally generated.  Okay, so SLNZ is the lattice in SLNR, hence has t.  Okay, so Kashtang's property t is inherited by quotients and is preserved undertaking  finite index subgroups.  If a group G acts fixed on freely on a tree, then G does not satisfy property t.  So this connection between property t and CS property of A was proven independently by  Alperin and Vatatani.  So trees are one-dimensional KDL cube complexes.  And nebulant trees generalize this factor higher-dimensional cube complexes.  So they proved if a group G acts fixed on freely on a KDL cube complex, then G does  not satisfy property t.  And this fact was recently used by Osaida to show that burnt-side groups do not have  t.  So we are mostly interested in discrete groups of property t and we have the following motivation  to search for a discrete t group.  So if a nice discrete group G has t, then one can construct a family of really nice  graphs, so a family of expander using calligraphs of finite quotients.  So I like groups and graphs and other geometric objects and I like to construct nice geometric  objects using groups.  And this fact is enough motivation for me to search for t groups.  So free groups and free Abellion groups do not have property t, but the automaphy's group  offer free group has t and the automaphy's group offer free Abellion group has t.  So coxsider groups are well understood objects in geometric group theory.  So they provide interesting examples and counter examples to many conjectures.  And I think it is natural to ask if the automaphy's group of a nice coxsider group has t.  So we conjecture that this is not the case.  So the automaphy's group of an infinite coxsider group does not have Kashtan's property t.  So some results in this direction will let w gamma be an infinite coxsider group.  So if gamma is discrete, so if gamma has no edges, then out w gamma does not have property  t.  And this result holds for all right angled coxsider groups.  So if w gamma is a right angled one, then the automaphy's group does not have property  t.  Okay, so let's talk about methods to prove that a group does not have t.  So if the outer automaphy's group is finite, then the automaphy's group of the coxsider  group does not have t.  So why?  So this follows from the fact that the center of a coxsider group is always finite and the  quotient w gamma, center of w gamma does not have t.  So it was proven by Howlett, Rowley and Taylor that if gamma is complete, then the outer  automaphy's group of the coxsider group is finite, hence the automaphy's group has no  t.  Okay, so Nibbler and Reeves constructed for each coxsider group a kazero cube complex  and which the group acts fixed on freely.  So what can we do with this complex?  So we can try to extend the action of the coxsider group w gamma on the corresponding  Nibbler-Reeves-Kazero-Cube complex to an action of a finite index subgroup of out w gamma.  So for some coxsider groups, this is doable, but for a general coxsider group, we don't  even know if the automaphy's group is finitely generated or not.  Okay, so we can try to show that the automaphy's group maps virtually onto w gamma or onto  a special infinite subgroup of w gamma, and this is what we are going to do.  So let me state the main theorem.  So let w gamma be an infinite coxsider group.  If gamma has a maximal complete subgraph delta such that the center of w delta is trivial,  so this assumption can be checked easily on the graph gamma.  So then the automaphy's group virtually subjects onto w gamma.  And in that case, the automaphy's group does not have t.  Okay, the second result is as follows.  So let w gamma be an infinite coxsider group.  If there exist two non-adjacent even vertices, vw and gamma, then the automaphy's group  virtually subjects onto a full product of the two.  So in particular, no t for the automaphy's group.  And what do I mean by an even vertex?  So a vertex, v in v, is called even if all h labels of e in e with v in e are even.  So let us discuss some examples.  Okay, so the corresponding coxsider group of this graph is the right angled one and  it is infinite.  So hence the group has no t.  So what about this graph?  Okay, so this graph has two non-adjacent even vertices.  So this vertex is an even one and this vertex is an even one.  Hence the group out w gamma has no t by part two of the main theorem.  Next example, so this graph has only one even vertex, this one.  So we cannot apply part two of the main theorem, but this graph has a maximum complete subgraph  data such that w data is an irreducible infinite coxsider group.  So let us consider this triangle, so triangle is the maximum complete subgraph in the graph  gamma.  The corresponding special coxsider subgroup is an alpha and one of type A2 tilde.  It is irreducible and irreducible coxsider groups have all this trivial center.  Hence the automorphism has no t by part one of the main theorem.  Okay, so more results.  So let w gamma be an infinite coxsider group again.  So if gamma has no cycle of length three, then the automorphism has no t.  So why?  Okay, so in this case, maximum complete subgraphs in gamma are edges.  And if gamma has an edge where the edge label is odd, then the center of this corresponding  special subgroup is trivial and we can apply part one of the main theorem to see that the  automorphism group has no t.  So otherwise, all edge labels are even and by part two of the main theorem follows that  the automorphism group has no t.  Okay, so if all edge labels are even, then the automorphism group does not have property  t.  So why?  So if gamma is complete, then the outer automorphism group is finite.  Hence the automorphism group has no t.  And if gamma is not complete, then gamma has two non-adjacent even vertices, hence by part  two of the main theorem, the automorphism group does not satisfy property t.  So if all edge labels are at least three, then out w gamma does not have property t.  So it follows from a characterization of finite coxseta groups that if gamma has at least  three vertices and all edge labels are at least three, then all special subgroups of  w gamma are infinite and irreducible.  So all non-trivial special subgroups, so all infinite special subgroups of w gamma are  irreducible.  So irreducible infinite coxseta groups have all the three real center.  So in particular, gamma has a maximal complete subgraph such that the corresponding special  subgroup has three real center.  Hence by part one of the main theorem, the automorphism group has no t.  Okay, so one more example.  So what about this graph?  Okay, so this graph has two connected components and has no even vertices.  So maximal complete subgraphs are triangles.  So we have eight triangles and the corresponding special subgroup of a triangle has non-trivial  center.  So we cannot apply the results in the main theorem to this automorphism group.  So question, does out w gamma satisfy property t?  We don't know.  Okay, so back to the main theorem.  And today I want to discuss the main steps in the proof of part one.  Okay, so an important puzzle piece of the proof is the following lemma.  So let gamma be a coxseta graph and delta one to delta n, the maximal complete subgraph.  We define w delta i in brackets as the following set.  Then the following map is a well-defined homomorphism.  And in particular the kernel of phi has finite index in the automorphism group.  Okay, so the idea of the proof is based on the following fact.  Maximal FA subgroups of a coxseta group w gamma are exactly conjugates of special subgroups  w delta i.  And this result was proved by Mihalyk and Chanst.  So for w delta i, the group, so the image of w delta i under F is again a maximal FA  subgroup.  So we know that the existing J such that we have this relation.  So such that as image of w delta i under F is in this set.  And then one can prove that the image of F is indeed a projection and phi is a homomorphism.  Okay, one remark.  So for F in the kernel of phi, the existing A in w gamma such that the image of w delta  i under F is a conjugate of w delta i by A. And we can compose the map F with C A inverse  where C A inverse is a conjugation by A inverse.  And then C A inverse composed with F is an automorphism of w delta i.  Okay, so back to the theorem and let us discuss the main steps in the proof.  Okay, first we construct a map psi from kernel of phi to the outer automorphism group of  w delta.  And then we show that the kernel of psi maps onto w gamma.  So the kernel of psi has finite index in the auto, the automorphism group of w gamma and  w gamma does not satisfy property t.  Since the group out w gamma does not have property t neither.  Okay, so let us discuss the main step is a proof more precisely.  So we already know that for F in the kernel of psi, the existing A in w gamma such that  the image of w delta under F is a conjugate of w delta by A.  And then we set the image of F under psi equal C A inverse composed with F and we take the  conjugacy class where C A inverse is again conjugation by A inverse.  And this map is a well-defined homomorphism, we'll have to check it.  So then we construct a map psi from kernel of psi to w gamma.  So for F in the kernel of psi, the existing B in w gamma such that the image of w under  F is a conjugation by B or all w in w delta and we set psi F equal B.  Okay, so this map is a well-defined one.  So why?  Because this element B is unique with this property.  Okay, so assume that the existing C in w gamma such that the image of w under F is a conjugation  by C.  And then B inverse C is in the centralizer of w delta in w gamma.  And this centralizer is equal to the center of w delta and the center of w delta is by  a sumptuous trivial.  So B is equal C and one can also prove that the map psi is an epimorphism.  Okay.  Okay, so recall a group G is called large if it virtually subjects onto the free group  of two generators.  So it is known that any coccister group is either virtually abelian or large.  So moreover, these properties can be verified easily on the coccister graph.  So concerning largeness of automorphism groups of coccister groups, so it was proven by Zalle  and Zuse that the auto-automorphism group of a right angled coccister group is either  large or virtually abelian.  So we obtain the following result concerning largeness of automorphism groups.  So let's w gamma be a large coccister group.  If gamma has a maximum complete subgraph delta such that the center of w delta is trivial,  then the group out w gamma is large.  Really nice result.  So what's next?  A very interesting class of groups is the class consisting of right angled arting groups.  So let us give a quick definition.  So given a finite simplisher of graph gamma, the right angled arting group A gamma is defined  as follows.  So the generator set is equal to the vertex set of the graph and for each edge in the  graph we have one relation.  So for each edge, uv, we have a relation uv, u minus 1, v to the power of minus 1.  So let us proceed with some examples regarding property T of automorphism group of a right  angled arting group.  So the group out of 5 has T. For n bigger or equal to 6, the group out of n has T, but  the group out AC5 where C5 is the pentagon does not have T. And there are more examples  in the literature.  I plan to answer the following question.  So for which shape of the graph gamma does the automorphism group out A gamma have Kashtan's  property T?  So I have results for small graphs, but these results will be the topic for my next talk.  So thank you for your attention.
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