Embedding Relatively Hyperbolic Groups in Products of Trees
نویسندگان
چکیده
We show that a relatively hyperbolic group quasiisometrically embeds in a product of finitely many trees if the peripheral subgroups do, and we provide an estimate on the minimal number of trees needed. Applying our result to the case of 3manifolds, we show that fundamental groups of closed 3-manifolds have linearly controlled asymptotic dimension at most 8. To complement this result, we observe that fundamental groups of Haken 3-manifolds with non-empty boundary have asymptotic dimension 2.
منابع مشابه
Embedding of hyperbolic Coxeter groups into products of binary trees and aperiodic tilings
We prove that a finitely generated, right-angled, hyperbolic Coxeter group Γ can be quasiisometrically embedded into the product of n binary trees, where n is the chromatic number of Γ. As application we obtain certain strongly aperiodic tilings of the Davis complex of these groups.
متن کاملEmbedding of Coxeter groups in a product of trees
We prove that a right angled Coxeter group Γ with chromatic number n can be embedded in a bilipschitz way into the product of n locally finite trees. We give applications of this result to various embedding problems and determine the hyperbolic rank of products of exponentially branching trees.
متن کاملRelative Hyperbolicity , Trees of Spaces And
We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalises a result of Bowditch for punctured su...
متن کاملBounded geometry in relatively hyperbolic groups
If a group is relatively hyperbolic, the parabolic subgroups are virtually nilpotent if and only if there exists a hyperbolic space with bounded geometry on which it acts geometrically finitely. This provides, via the embedding theorem of M. Bonk and O. Schramm, a very short proof of the finiteness of asymptotic dimension for such groups (which is known to imply Novikov conjectures).
متن کاملA Combination Theorem for Strong Relative Hyperbolicity
We prove a combination theorem for trees of (strongly) relatively hyperbolic spaces and finite graphs of (strongly) relatively hyperbolic groups. This gives a geometric extension of Bestvina and Feighn’s Combination Theorem for hyperbolic groups and answers a question of Swarup. We also prove a converse to the main Combination Theorem. AMS subject classification = 20F32(Primary), 57M50(Secondary)
متن کامل