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Search: MSC category 20F67 ( Hyperbolic groups and nonpositively curved groups )

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1. CMB Online first

Button, Jack
Tubular free by cyclic groups act freely on CAT(0) cube complexes
We identify when a tubular group (the fundamental group of a finite graph of groups with $\mathbb{Z}^2$ vertex and $\mathbb{Z}$ edge groups) is free by cyclic and show, using Wise's equitable sets criterion, that every tubular free by cyclic group acts freely on a CAT(0) cube complex.

Keywords:CAT(0), tubular group
Categories:20F65, 20F67, 20E08

2. CMB 2016 (vol 59 pp. 244)

Cao, Wensheng; Huang, Xiaolin
A Note on Quaternionic Hyperbolic Ideal Triangle Groups
In this paper, the quaternionic hyperbolic ideal triangle groups are parameterized by a real one-parameter family $\{\phi_s: s\in \mathbb{R}\}$. The indexing parameter $s$ is the tangent of the quaternionic angular invariant of a triple of points in $\partial \mathbf{H}_{\mathbb{h}}^2 $ forming this ideal triangle. We show that if $s \gt \sqrt{125/3}$ then $\phi_s$ is not a discrete embedding, and if $s \leq \sqrt{35}$ then $\phi_s$ is a discrete embedding.

Keywords:quaternionic inversion, ideal triangle group, quaternionic Cartan angular invariant
Categories:20F67, 22E40, 30F40

3. CMB 2015 (vol 59 pp. 170)

Martínez-Pedroza, Eduardo
A Note on Fine Graphs and Homological Isoperimetric Inequalities
In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected $2$-complex $X$ with a linear homological isoperimetric inequality, a bound on the length of attaching maps of $2$-cells and finitely many $2$-cells adjacent to any edge must have a fine $1$-skeleton. We provide a positive answer to this question. We revisit a homological characterization of relative hyperbolicity, and show that a group $G$ is hyperbolic relative to a collection of subgroups $\mathcal P$ if and only if $G$ acts cocompactly with finite edge stabilizers on an connected $2$-dimensional cell complex with a linear homological isoperimetric inequality and $\mathcal P$ is a collection of representatives of conjugacy classes of vertex stabilizers.

Keywords:isoperimetric functions, Dehn functions, hyperbolic groups
Categories:20F67, 05C10, 20J05, 57M60

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