On growth of double cosets in hyperbolic groups

Let [Formula: see text] be a hyperbolic group, [Formula: see text] and [Formula: see text] be subgroups of [Formula: see text], and [Formula: see text] be the growth function of the double cosets [Formula: see text]. We prove that the behavior of [Formula: see text] splits into two different cases. If [Formula: see text] and [Formula: see text] are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as [Formula: see text]. We can even take [Formula: see text]. In contrast, for quasiconvex subgroups [Formula: see text] and [Formula: see text] of infinite index, [Formula: see text] is exponential. Moreover, there exists a constant [Formula: see text], such that [Formula: see text] for all big enough [Formula: see text], where [Formula: see text] is the growth function of the group [Formula: see text]. So, we have a clear dichotomy between the quasiconvex and non-quasiconvex case.

Growth of quasiconvex subgroups

We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron–Frobenius theory.We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions use the notion of growth tightness and the work of Dahmani, Guirardel and Osin on rotating families.

On intersections of conjugate subgroups

We define a new invariant of a conjugacy class of subgroups which we call the breadth and prove that a quasiconvex subgroup of a negatively curved group has finite breadth in the ambient group. Utilizing the coset graph and the geodesic core of a subgroup we give an explicit algorithm for constructing a finite generating set for an intersection of a quasiconvex subgroup of a negatively curved group with its conjugate. Using that algorithm we construct algorithms for computing the breadth, the width, and the height of a quasiconvex subgroup of a negatively curved group. These algorithms decide if a quasiconvex subgroup of a negatively curved group is almost malnormal in the ambient group. We also explicitly compute a quasiconvexity constant of the intersection of two quasiconvex subgroups and give examples demonstrating that height, width, and breadth are different invariants of a subgroup.

On convex hulls and the quasiconvex subgroups of Fm×ℤn

In this paper, we explore a method for forming the convex hull of a subset in a uniquely geodesic metric space due to Brunn and use it to show that with respect to the usual action of