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Anne Thomas

Publications and source records attributed to Anne Thomas.

At least 37 records · Page 2Linked to original sources
arXiv2016

Maximal torsion-free subgroups of certain lattices of hyperbolic buildings and Davis complexes

We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and construct a finite-index embedding of the fundamental group into the Coxeter group whose Davis complex is the universal cover of the polygonal complex. All of the groups which we embed have minimal index among torsion-free subgroups, and therefore are maximal among torsion-free subgroups.

math.GR↗
arXiv2016

C*-algebras associated to graphs of groups

To a large class of graphs of groups we associate a C*-algebra universal for generators and relations. We show that this C*-algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of groups on the boundary of its Bass-Serre tree. We characterise when this action is minimal, and find a sufficient condition under which it is locally contractive. In the case of generalised Baumslag-Solitar graphs of groups (graphs of groups in which every group is infinite cyclic) we also characterise topological freeness of this action. We are then able to establish a dichotomy for simple C*-algebras associated to generalised Baumslag-Solitar graphs of groups: they are either a Kirchberg algebra, or a stable Bunce-Deddens algebra.

math.OA↗
arXiv2014

Infinite reduced words and the Tits boundary of a Coxeter group

Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special cases, including W word hyperbolic, and X with isolated flats. We establish that when W is word hyperbolic, the limit weak order is the disjoint union of weak orders of finite Coxeter groups. We also establish, for each boundary point ξ, a natural order-preserving correspondence between infinite reduced words which "point towards" ξ, and elements of the reflection subgroup of W which fixes ξ.

math.GR↗
arXiv2013

Divergence in right-angled Coxeter groups

Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex.

math.GR↗
arXiv2013

Characterising star-transitive and st(edge)-transitive graphs

Recent work of Lazarovich provides necessary and sufficient conditions on a graph L for there to exist a unique simply-connected (k,L)-complex. The two conditions are symmetry properties of the graph, namely star-transitivity and st(edge)-transitivity. In this paper we investigate star-transitive and st(edge)-transitive graphs by studying the structure of the vertex and edge stabilisers of such graphs. We also provide new examples of graphs that are both star-transitive and st(edge)-transitive.

math.CO↗
arXiv2012

Cocompact lattices in complete Kac-Moody groups with Weyl group right-angled or a free product of spherical special subgroups

Let G be a complete Kac-Moody group of rank n \geq 2 over the finite field of order q, with Weyl group W and building Δ. We first show that if W is right-angled, then for all q \neq 1 mod 4 the group G admits a cocompact lattice Γwhich acts transitively on the chambers of Δ. We also obtain a cocompact lattice for q =1 mod 4 in the case that Δis Bourdon's building. As a corollary of our constructions, for certain right-angled W and certain q, the lattice Γhas a surface subgroup. We also show that if W is a free product of spherical special subgroups, then for all q, the group G admits a cocompact lattice Γwith Γa finitely generated free group. Our proofs use generalisations of our results in rank 2 concerning the action of certain finite subgroups of G on Δ, together with covering theory for complexes of groups.

math.GR↗
arXiv2012

Cocompact lattices on \tilde{A}_n buildings

Let K be the field of formal Laurent series over the finite field of order q. We construct cocompact lattices Γ'_0 < Γ_0 in the group G = PGL_d(K) which are type-preserving and act transitively on the set of vertices of each type in the building associated to G. The stabiliser of each vertex in Γ'_0 is a Singer cycle and the stabiliser of each vertex in Γ_0 is isomorphic to the normaliser of a Singer cycle in PGL_d(q). We then show that the intersections of Γ'_0 and Γ_0 with PSL_d(K) are lattices in PSL_d(K), and identify the pairs (d,q) such that the entire lattice Γ'_0 or Γ_0 is contained in PSL_d(K). Finally we discuss minimality of covolumes of cocompact lattices in SL_3(K). Our proofs combine a construction of Cartwright and Steger with results about Singer cycles and their normalisers, and geometric arguments.

math.GR↗
arXiv2012

Density of commensurators for uniform lattices of right-angled buildings

Let G be the automorphism group of a regular right-angled building X. The "standard uniform lattice" Γ_0 in G is a canonical graph product of finite groups, which acts discretely on X with quotient a chamber. We prove that the commensurator of Γ_0 is dense in G. This result was also obtained by Haglund. For our proof, we develop carefully a technique of "unfoldings" of complexes of groups. We use unfoldings to construct a sequence of uniform lattices Γ_n in G, each commensurable to Γ_0, and then apply the theory of group actions on complexes of groups to the sequence Γ_n. As further applications of unfoldings, we determine exactly when the group G is nondiscrete, and we prove that G acts strongly transitively on X.

math.GR↗
arXiv2012

Lattices in hyperbolic buildings

This survey is a brief introduction to the theory of hyperbolic buildings and their lattices, with a focus on recent results.

math.GR↗
arXiv2011

Lattices in complete rank 2 Kac-Moody groups

Let Λbe a minimal Kac-Moody group of rank 2 defined over the finite field F_q, where q = p^a with p prime. Let G be the topological Kac-Moody group obtained by completing Λ. An example is G=SL_2(K), where K is the field of formal Laurent series over F_q. The group G acts on its Bruhat-Tits building X, a tree, with quotient a single edge. We construct new examples of cocompact lattices in G, many of them edge-transitive. We then show that if cocompact lattices in G do not contain p-elements, the lattices we construct are the only edge-transitive lattices in G, and that our constructions include the cocompact lattice of minimal covolume in G. We also observe that, with an additional assumption on p-elements in G, the arguments of Lubotzky for the case G = SL_2(K) may be generalised to show that there is a positive lower bound on the covolumes of all lattices in G, and that this minimum is realised by a non-cocompact lattice, a maximal parabolic subgroup of Lambda.

math.GR↗
arXiv2011

Cocompact lattices of minimal covolume in rank 2 Kac-Moody groups, Part II

Let G be a topological Kac-Moody group of rank 2 with symmetric Cartan matrix, defined over a finite field F_q. An example is G = SL(2,F_q((t^{-1}))). We determine a positive lower bound on the covolumes of cocompact lattices in G, and construct a cocompact lattice Γ_0 < G which realises this minimum. This completes the work begun in Part I, which considered the cases when G admits an edge-transitive lattice.

math.GR↗
arXiv2011

Existence, covolumes and infinite generation of lattices for Davis complexes

Let $Σ$ be the Davis complex for a Coxeter system (W,S). The automorphism group G of $Σ$ is naturally a locally compact group, and a simple combinatorial condition due to Haglund--Paulin determines when G is nondiscrete. The Coxeter group W may be regarded as a uniform lattice in G. We show that many such G also admit a nonuniform lattice $Γ$, and an infinite family of uniform lattices with covolumes converging to that of $Γ$. It follows that the set of covolumes of lattices in G is nondiscrete. We also show that the nonuniform lattice $Γ$ is not finitely generated. Examples of $Σ$ to which our results apply include buildings and non-buildings, and many complexes of dimension greater than 2. To prove these results, we introduce a new tool, that of "group actions on complexes of groups", and use this to construct our lattices as fundamental groups of complexes of groups with universal cover $Σ$.

math.GR↗
arXiv2011

Surface quotients of hyperbolic buildings

Let I(p,v) be Bourdon's building, the unique simply-connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons and the link at each vertex is the complete bipartite graph K(v,v). We investigate and mostly determine the set of triples (p,v,g) for which there exists a uniform lattice Γ in Aut(I(p,v)) such that Γ\I(p,v) is a compact orientable surface of genus g. Surprisingly, the existence of Γ depends upon the value of v. The remaining cases lead to open questions in tessellations of surfaces and in number theory. Our construction of Γ, together with a theorem of Haglund, implies that for p>=6, every uniform lattice in Aut(I) contains a surface subgroup. We use elementary group theory, combinatorics, algebraic topology, and number theory.

math.GR↗
arXiv2009

Lattices acting on right-angled buildings

Let X be a right-angled building. We show that the lattices in Aut(X) share many properties with tree lattices. For example, we characterise the set of covolumes of uniform and of nonuniform lattices in Aut(X), and show that the group Aut(X) admits an infinite ascending tower of uniform and of nonuniform lattices. These results are proved by constructing a functor from graphs of groups to complexes of groups.

math.GR↗
arXiv2007

Hyperbolic Geometry and Distance Functions on Discrete Groups

Chapter 1 is a short history of non-Euclidean geometry, which synthesises my readings of mostly secondary sources. Chapter 2 presents each of the main models of hyperbolic geometry, and describes the tesselation of the upper half-plane induced by the action of $PSL(2,\mathbb{Z})$. Chapter 3 gives background on symmetric spaces and word metrics. Chapter 4 then contains a careful proof of the following theorem of Lubotzky--Mozes--Raghunathan: the word metric on $PSL(2,\mathbb{Z})$ is not Lipschitz equivalent to the metric induced by its action on the associated symmetric space (the upper half-plane), but for $n \geq 3$, these two metrics on $PSL(n,\mathbb{Z})$ are Lipschitz equivalent.

math.GR↗