Infinite generation of non-cocompact lattices on right-angled buildings

Thomas, Anne; Wortman, Kevin

doi:10.2140/agt.2011.11.929

Cited by 9 publications

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“…As a corollary, such lattices are not finitely presentable. The examples of [57] and [58] show that Gandini's bound is not sharp. 2.2.5.…”

Section: Property (T) and Finiteness Properties Ballmann-świ ֒mentioning

confidence: 96%

“…All cocompact lattices in a hyperbolic building are finitely generated since they are fundamental groups of finite complexes of finite groups. Infinite generation of some non-cocompact lattices for certain hyperbolic buildings was established by Thomas [57] and Thomas-Wortman [58]. It is not known whether for these buildings, there are any non-cocompact lattices which are finitely generated.…”

Section: Property (T) and Finiteness Properties Ballmann-świ ֒mentioning

confidence: 99%

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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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“…As a corollary, such lattices are not finitely presentable. The examples of [57] and [58] show that Gandini's bound is not sharp. 2.2.5.…”

Section: Property (T) and Finiteness Properties Ballmann-świ ֒mentioning

confidence: 96%

Section: Property (T) and Finiteness Properties Ballmann-świ ֒mentioning

confidence: 99%

Lattices in hyperbolic buildings

Thomas¹

2016

Geometry, Topology, and Dynamics in Negative Curvature

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“…We want to describe the equivalence classes of parallelism of panels in rightangled buildings. It turns out that these classes are the so called tree-walls, initially defined in [Bou97] and taken over in [TW11]. Each tree-wall separates the building into combinatorially convex components, which will be called wings as in [Cap14].…”

Section: Tree-walls and Wingsmentioning

confidence: 99%

“…(Such a result is not true at all for buildings in general.) A more systematic study of combinatorial properties of right-angled buildings was initiated by Anne Thomas and Kevin Wortman in [TW11] and continued by Pierre-Emmanuel Caprace in [Cap14], and we rely on their results.…”

Section: Introductionmentioning

confidence: 99%

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Universal groups for right-angled buildings

2018

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In 2000, M. Burger and S. Mozes introduced universal groups acting on trees with a prescribed local action. We generalize this concept to groups acting on right-angled buildings. When the right-angled building is thick and irreducible of rank at least 2 and each of the local permutation groups is transitive and generated by its point stabilizers, we show that the corresponding universal group is a simple group.When the building is locally finite, these universal groups are compactly generated totally disconnected locally compact groups, and we describe the structure of the maximal compact open subgroups of the universal groups as a limit of generalized wreath products.

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Automorphism Groups of Combinatorial Structures

Thomas

2018

MATRIX Book Series

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Infinite generation of non-cocompact lattices on right-angled buildings

Abstract: Let be a non-cocompact lattice on a locally finite regular right-angled building X . We prove that if has a strict fundamental domain then is not finitely generated. We use the separation properties of subcomplexes of X called tree-walls.20E42, 20F05; 20F55, 57M07, 51E24

Cited by 9 publications

References 12 publications

Lattices in hyperbolic buildings

Lattices in hyperbolic buildings

Universal groups for right-angled buildings

Automorphism Groups of Combinatorial Structures

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