Relative hyperbolicity for automorphisms of free products and free groups

We prove that for a free product [Formula: see text] with free factor system [Formula: see text], any automorphism [Formula: see text] preserving [Formula: see text], atoroidal (in a sense relative to [Formula: see text]) and none of whose power send two different conjugates of subgroups in [Formula: see text] on conjugates of themselves by the same element, gives rise to a semidirect product [Formula: see text] that is relatively hyperbolic with respect to suspensions of groups in [Formula: see text]. We recover a theorem of Gautero–Lustig and Ghosh that, if [Formula: see text] is a free group, [Formula: see text] an automorphism of [Formula: see text], and [Formula: see text] is its family of polynomially growing subgroups, then the semidirect product by [Formula: see text] is relatively hyperbolic with respect to the suspensions of these subgroups. We apply the first result to the conjugacy problem for certain automorphisms (atoroidal and toral) of free products of abelian groups.

From Hierarchical to Relative Hyperbolicity

We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our criteria to graphs associated to surfaces and prove that the separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures. We also recover a celebrated theorem of Brock and Masur on the relative hyperbolicity of the Weil–Petersson metric on Teichmüller space for surfaces with complexity three.

The relative hyperbolicity and manifold structure of certain right-angled Coxeter groups

In this paper, we study the manifold structure and the relative hyperbolic structure of right-angled Coxeter groups with planar nerves. We then apply our results to the quasi-isometry problem for this class of right-angled Coxeter groups.

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 attachingmaps 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 P if and only if G acts cocompactly with ûnite edge stabilizers on a connected 2-dimensional cell complex with a linear homological isoperimetric inequality and P is a collection of representatives of conjugacy classes of vertex stabilizers.

The rapid decay property and centroids in groups

This is a survey of methods of proving or disproving the rapid decay property in groups. We present a centroid property of group actions on metric spaces. That property is a generalized (and corrected) version of the "(**)-relative hyperbolicity" from [9] and implies the rapid decay (RD) property. We show that several properties which are known to imply RD also imply the centroid property. Thus uniform lattices in many semi-simple Lie groups, graph products of groups, Artin groups of large type and the mapping class groups have the (relative) centroid property. We also present a simple "non-amenability-like" property that follows from RD, and give an easy example of a group without RD and without any amenable subgroup with superpolynomial growth.