Density of commensurators for uniform lattices of right-angled buildings

Kubena, Angela; Thomas, Anne

doi:10.1515/jgt-2012-0017

Cited by 8 publications

(9 citation statements)

References 27 publications

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“…The commensurator of a lattice Γ < G is the subgroup consisting of elements g ∈ G such that gΓg −1 and Γ are commensurable. Haglund [34] and independently Kubena-Thomas [43] proved that for ∆ a regular right-angled building, the commensurator of a canonical cocompact lattice is dense in G = Aut(∆). The question of commensurators of non-cocompact lattices is wide open, even for I p,q .…”

Section: 21mentioning

confidence: 99%

Lattices in hyperbolic buildings

Thomas¹

2016

Geometry, Topology, and Dynamics in Negative Curvature

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1.1. Coxeter groups and Coxeter polytopes. We mostly follow the reference Davis [19], particularly Chapter 6, and concentrate on the hyperbolic case.Recall that a Coxeter group is a group W with finite generating set S and presentation of the form W = s ∈ S | (st) mst = 1 where s, t ∈ S, m ss = 1 for all s ∈ S and if s = t then m st is an integer ≥ 2 or m st = ∞, meaning that the product st has infinite order. The pair (W, S) is called a Coxeter system. A Coxeter system (W, S) is right-angled if for each s, t ∈ S with s = t, m st ∈ {2, ∞}. Note that m st = 2 if and only if st = ts. Let X n be the n-dimensional sphere, n-dimensional Euclidean space or n-dimensional (real) hyperbolic space. Many important examples of Coxeter groups arise as discrete reflection groups acting on X n , as follows. Let P be a convex polyhedron in X n with all dihedral angles integer submultiples of π. Such a P is called a Coxeter polytope. Let W = W (P ) be the group generated by the set S = S(P ) of reflections in the codimension one faces of P . Then (W, S) is a Coxeter system and W is a discrete subgroup of the isometry group of X n (see [19, Theorem 6.4.3]). Moreover, the action of W tessellates X n by copies of P . For example, let P be a right-angled hyperbolic p-gon, p ≥ 5. Then the corresponding Coxeter system is right-angled with p generators, one for each side of P , so that if s and t are reflections in distinct sides then m st = 2 when these sides are adjacent, and otherwise m st = ∞.Suppose that X n is the sphere or Euclidean space. Then Coxeter polytopes P ⊂ X n exist and have been classified in every dimension, and the corresponding Coxeter systems (W, S) are the spherical or affine Coxeter systems, respectively (see [19, Table 6.1] for the classification).If X n is n-dimensional hyperbolic space H n , then there is no complete classification of Coxeter polytopes. Vinberg's Theorem [65] establishes that compact hyperbolic Coxeter polytopes can exist only in dimension n ≤ 29, although at the time of writing the highest dimension in which an example is known is n = 8 (due to Bugaenko [12]). Finite volume hyperbolic Coxeter polytopes have also been investigated, with for example Prokhorov [50] proving these can exist only in dimension n ≤ 995. For n ≤ 6 (respectively, n ≤ 19), there are infinitely many essentially distinct compact (respectively, finite volume) hyperbolic Coxeter polytopes (Allcock [2]). In dimension 3, Andreev's Theorem [3] classifies compact hyperbolic Coxeter polytopes, but in dimensions n ≥ 4 only special cases have been considered, and there seems little hope of a complete list. Hyperbolic Coxeter polytopes which are simplices exist in dimensions n ≤ 4 only, and their classification is given in [19, Table 6.2]. Right-angled compact hyperbolic polytopes also exist in dimensions n ≤ 4 only, and there are infinitely many examples in each dimension n ≤ 4 (see [66]). A right-angled example in dimension 3 is the dodecahedron, which tessellates H 3 as depicted on the cover of Thurston's book [59], and a right-...

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Section: 21mentioning

confidence: 99%

Lattices in hyperbolic buildings

Thomas¹

2016

Geometry, Topology, and Dynamics in Negative Curvature

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“…We finally note that since all local groups of G(Y ) are direct products of cyclic groups, it seems possible that all of its nontrivial local groups (not just the nontrivial face groups) could be killed using a construction similar to that in [1] or [15], so as to obtain the infinite polygonal complex Y g = π 1 (S g )\I p,v explicitly.…”

Section: Relationships With Previous Examples and Surface Subgroupsmentioning

confidence: 99%

“…In this topology, a uniform lattice in G is a subgroup Γ < G acting cocompactly on I p,v with finite cell stabilizers (see Section 2.2 below). Bourdon's building and its lattices have been studied by, for example, Bourdon [4], Bourdon-Pajot [5], Haglund [9,10,11], Kubena-Thomas [15], Ledrappier-Lim [16], Rémy [18], Thomas [20], and Vdovina [21].…”

Section: Introductionmentioning

confidence: 99%

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Surface Quotients of Hyperbolic Buildings

Futer

Thomas

2011

International Mathematics Research Notices

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Abstract. Let Ip,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 Kv,v. We investigate and mostly determine the set of triples (p, v, g) for which there exists a uniform lattice Γ = Γp,v,g in Aut(Ip,v) such that Γ\Ip,v is a compact orientable surface of genus g. Surprisingly, for some p and g the existence of Γp,v,g depends upon the value of v. The remaining cases lead to open questions in tessellations of surfaces and in number theory. Our construction of Γp,v,g as the fundamental group of a simple complex of groups, together with a theorem of Haglund, implies that for p ≥ 6 every uniform lattice in Aut(Ip,v) contains a surface subgroup. We use elementary group theory, combinatorics, algebraic topology, and number theory.

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“…Let us denote the group of commensurators of Γ by Comm(Γ). Commensurator subgroup Comm(Γ) plays an important role in the study of rigidity of locally symmetric spaces and more generally in geometric group theory ( [2][3][4]).…”

Section: Introductionmentioning

confidence: 99%