arXiv subjects

David Hume

Publications and source records attributed to David Hume.

24 records · Page 2Linked to original sources

arXiv2017

Embedding mapping class groups into finite products of trees

We prove the equivalence between a relative bottleneck property and being quasi-isometric to a tree-graded space. As a consequence, we deduce that the quasi-trees of spaces defined axiomatically by Bestvina-Bromberg-Fujiwara are quasi-isometric to tree-graded spaces. Using this we prove that mapping class groups quasi-isometrically embed into a finite product of simplicial trees. In particular, these groups have finite Assouad-Nagata dimension, direct embeddings exhibiting $\ell^p$ compression exponent $1$ for all $p\geq 1$ and they quasi-isometrically embed into $\ell^1(\mathbb N)$. We deduce similar consequences for relatively hyperbolic groups whose parabolic subgroups satisfy such conditions. In obtaining these results we also demonstrate that curve complexes of compact surfaces and coned-off graphs of relatively hyperbolic groups admit quasi-isometric embeddings into finite products of trees.

math.MG↗

arXiv2016

A continuum of expanders

A regular equivalence between two graphs $Γ,Γ'$ is a pair of uniformly proper Lipschitz maps $VΓ\to VΓ'$ and $VΓ'\to VΓ$. Using separation profiles we prove that there are $2^{\aleph_0}$ regular equivalence classes of expander graphs, and of finitely generated groups with a representative which isometrically contains expanders.

math.GR↗

arXiv2016

Relatively hyperbolic groups with fixed peripherals

We build quasi--isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors. We prove that, given any finite collection of finitely generated groups $\mathcal{H}$ each of which either has finite stable dimension or is non-relatively hyperbolic, there exist infinitely many quasi--isometry types of one--ended groups which are hyperbolic relative to $\mathcal{H}$. The groups are constructed using small cancellation theory over free products.

math.GR↗

arXiv2015

Orthogonal forms of Kac--Moody groups are acylindrically hyperbolic

We give sufficient conditions for a group acting on a geodesic metric space to be acylindrically hyperbolic and mention various applications to groups acting on CAT($0$) spaces. We prove that a group acting on an irreducible non-spherical non-affine building is acylindrically hyperbolic provided there is a chamber with finite stabiliser whose orbit contains an apartment. Finally, we show that the following classes of groups admit an action on a building with those properties: orthogonal forms of Kac--Moody groups over arbitrary fields, and irreducible graph products of arbitrary groups - recovering a result of Minasyan--Osin.

math.GR↗

arXiv2013

Direct embeddings of relatively hyperbolic groups with optimal $\ell^p$ compression exponent

We prove that for all $p>1$, every relatively hyperbolic group has $\ell^p$ compression exponent equal to the minimum of the exponents of its maximal parabolic subgroups. Moreover, this embedding can be explicitly described in terms of given embeddings of the maximal parabolic subgroups.

math.MG↗

arXiv2012

Embedding universal covers of graph manifolds in products of trees

We prove that the universal cover of any graph manifold quasi-isometrically embeds into a product of three trees. In particular we show that the Assouad-Nagata dimension of the universal cover of any closed graph manifold is 3, proving a conjecture of Smirnov.

math.GT↗