On the Convex Closure of the Graph of Modular Inversions

In this paper we give upper and lower bounds as well as a heuristic estimate on the number of vertices of the convex closure of the set $$G_n=\left\{(a,b) : a,b\in \Z, ab \equiv 1 \pmod{n}, 1\leq a,b\leq n-1\right\}.$$ The heuristic is based on an asymptotic formula of R\'{e}nyi and Sulanke. After describing two algorithms to determine the convex closure, we compare the numeric results with the heuristic estimate. The numeric results do not agree with the heuristic estimate -- there are some interesting peculiarities for which we provide a heuristic explanation. We then describe some numerical work on the convex closure of the graph of random quadratic and cubic polynomials over $\mathbb{Z}_n$. In this case the numeric results are in much closer agreement with the heuristic, which strongly suggests that the the curve $xy=1\pmod{n}$ is ``atypical''.Comment: 33 pages, 14 figure

A combination theorem for combinatorially non-positively curved complexes of hyperbolic groups

We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch's characterization of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvature, and which holds for large classes of complexes. As an application, we study the hyperbolicity of groups obtained by small cancellation over a graph of hyperbolic groups.Comment: final preprint version, to appear in Math. Proc. Cambridge Philos. So

Amenable hyperbolic groups

We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semi-regular trees acting doubly transitively on the set of ends. As an application, we show that the class of hyperbolic locally compact groups with a cusp-uniform non-uniform lattice, is very restricted.Comment: 41 pages, no figure. v2: revised version (minor changes

Deformation spaces of trees

Let G be a finitely generated group. Two simplicial G-trees are said to be in the same deformation space if they have the same elliptic subgroups (if H fixes a point in one tree, it also does in the other). Examples include Culler-Vogtmann's outer space, and spaces of JSJ decompositions. We discuss what features are common to trees in a given deformation space, how to pass from one tree to all other trees in its deformation space, and the topology of deformation spaces. In particular, we prove that all deformation spaces are contractible complexes.Comment: Update to published version. 43 page