Surface Quotients of Hyperbolic Buildings

Futer, David; Thomas, Anne

doi:10.1093/imrn/rnr028

Cited by 8 publications

(14 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, see [4] for surface subgroups of right-angled Artin groups or [2], where Calegari and Walker show, that a random group contains many quasiconvex surface subgroups. Existence of surface subgroups in right-angled hyperbolic buildings was shown in [5], and in hyperbolic buildings with 4-gonal apartments in [14]. Existence of surface subgroups in fundamental groups of higher-dimensional complexes is discussed, for example, in [6].…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic Triangular Buildings Without Periodic Planes of Genus 2

Kangaslampi

Vdovina

2016

Experimental Mathematics

View full text Add to dashboard Cite

We study surface subgroups of groups acting simply transitively on vertex sets of certain hyperbolic triangular buildings. The study is motivated by Gromov's famous surface subgroup question: Does every one-ended hyperbolic group contain a subgroup which is isomorphic to the fundamental group of a closed surface of genus at least 2? In [9] and [3] the authors constructed and classified all groups acting simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. These groups gave the first examples of cocompact lattices acting simply transitively on vertices of hyperbolic triangular Kac-Moody buildings that are not right-angled. Here we study surface subgroups of the 23 torsion free groups obtained in [9]. With the help of computer searches we show, that in most of the cases there are no periodic apartments invariant under the action of a genus two surface. The existence of such an action implies the existence of a surface subgroup, but it is not known, whether the existence of a surface subgroup implies the existence of a periodic apartment. These groups are the first candidates for groups that have no surface subgroups arising from periodic apartments.

show abstract

Section: Introductionmentioning

confidence: 99%

Hyperbolic Triangular Buildings Without Periodic Planes of Genus 2

Kangaslampi

Vdovina

2016

Experimental Mathematics

View full text Add to dashboard Cite

show abstract

“…Thus our results show that if the building associated to G is Bourdon's building, then for all q the Kac-Moody group G admits a cocompact lattice. Lattices in the full automorphism group of Bourdon's building have been studied by several authors (see, for instance, [12] and references therein).…”

Section: Theorem 1 Let G Be a Complete Kac-moody Group Of Rank N ≥ 2mentioning

confidence: 99%

“…, A i n are non-trivial, the fact that Γ has a surface subgroup follows from either Holt-Rees [15 Proof. The construction of Γ in Section 4.3 uses a simple complex of groups from Futer-Thomas [12], so that by the same arguments as in [12], the group Γ has a surface subgroup. Now suppose n ≥ 6 and that q is any power of a prime.…”

Section: Proof Of the Surface Subgroup Corollarymentioning

confidence: 99%

See 1 more Smart Citation

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

Capdeboscq

Thomas

2013

Mathematical Research Letters

Self Cite

View full text Add to dashboard Cite

Abstract. Let G be a complete Kac-Moody group of rank n ≥ 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 ≡ 1 (mod 4) the group G admits a cocompact lattice Γ which acts transitively on the chambers of Δ. We also obtain a cocompact lattice for q ≡ 1 (mod 4) in the case that Δ is Bourdon's building. As a corollary of our constructions, for certain rightangled W and certain q, the lattice Γ has a surface subgroup. Our second main result states that if W is a free product of spherical special subgroups, then for all q, the group G admits a cocompact lattice Γ, with Γ a finitely generated free group. Thus for many G of rank n ≥ 3, our results provide the first (explicit) constructions of cocompact lattices in G, and in particular, the first explicit examples of cocompact lattices in complete Kac-Moody groups whose Weyl group is not right-angled. Our proofs use generalizations of our results in rank 2 [5] concerning the action of certain finite subgroups of G on Δ, together with covering theory for complexes of groups.

show abstract

“…In fact this result was already known for n ≥ 5 (and it is straightforward to prove for n=3,4); however our proof is short and elementary. For n ≥ 5 the result is proved in Sang-hyun Kim's 2007 thesis [16,Theorem 3.6] using Van Kampen diagrams, and for n ≥ 6 another quite different proof is provided in [15,Corollary 1.3].…”

Section: Generalising Servatius Droms and Servatiusmentioning

confidence: 99%

Generalising some results about right-angled Artin groups to graph products of groups

Holt

Rees

2012

Journal of Algebra

View full text Add to dashboard Cite

We prove three results about the graph product G = G(Γ; G v , v ∈ V (Γ)) of groups G v over a graph Γ. The first result generalises a result of Servatius, Droms and Servatius, proved by them for rightangled Artin groups; we prove a necessary and sufficient condition on a finite graph Γ for the kernel of the map from G to the associated direct product to be free (one part of this result already follows from a result in S. Kim's Ph.D. thesis). The second result generalises a result of Hermiller andSunić, again from right-angled Artin groups; we prove that, for a graph Γ with finite chromatic number, G has a series in which every factor is a free product of vertex groups. The third result provides an alternative proof of a theorem due to Meier, which provides necessary and sufficient conditions on a finite graph Γ for G to be hyperbolic.

show abstract

Surface Quotients of Hyperbolic Buildings

Cited by 8 publications

References 17 publications

Hyperbolic Triangular Buildings Without Periodic Planes of Genus 2

Hyperbolic Triangular Buildings Without Periodic Planes of Genus 2

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

Generalising some results about right-angled Artin groups to graph products of groups

Contact Info

Product

Resources

About