2,735 research outputs found
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.Comment: 3 pages, final version - to appear in Proceedings of the AM
Poorly connected groups
We investigate groups whose Cayley graphs have poor\-ly connected subgraphs.
We prove that a finitely generated group has bounded separation in the sense of
Benjamini--Schramm--Tim\'ar if and only if it is virtually free. We then prove
a gap theorem for connectivity of finitely presented groups, and prove that
there is no comparable theorem for all finitely generated groups. Finally, we
formulate a connectivity version of the conjecture that every group of type
with no Baumslag-Solitar subgroup is hyperbolic, and prove it for groups with
at most quadratic Dehn function.Comment: 14 pages. Changes to v2: Proof of the Theorem 1.2 shortened, Theorem
1.4 added completing the no-gap result outlined in v
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() 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.Comment: 20 pages, to appear in Annales de l'Institut Fourie
Poincar\'e profiles of groups and spaces
We introduce a spectrum of monotone coarse invariants for metric measure
spaces called Poincar\'{e} profiles. The two extremes of this spectrum
determine the growth of the space, and the separation profile as defined by
Benjamini--Schramm--Tim\'{a}r. In this paper we focus on properties of the
Poincar\'{e} profiles of groups with polynomial growth, and of hyperbolic
spaces, where we deduce a connection between these profiles and conformal
dimension. As applications, we use these invariants to show the non-existence
of coarse embeddings in a variety of examples.Comment: 55 pages. To appear in Revista Matem\'atica Iberoamerican
A continuum of expanders
A regular equivalence between two graphs is a pair of
uniformly proper Lipschitz maps and . Using separation profiles we prove that there are
regular equivalence classes of expander graphs, and of finitely generated
groups with a representative which isometrically contains expanders.Comment: 11 pages, accepted for publication in Fundamenta Mathematica
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