Related papers: Lower bound on growth of non-elementary subgroups …

Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a…

We establish growth tightness for a class of groups acting geometrically on a geodesic metric space and containing a contracting element. As a consequence, any group with nontrivial Floyd boundary are proven to be growth tight with respect…

The following short note provides an alternative proof of a result of Coornaert: namely, that given a non-elementary word-hyperbolic group $G$ with a finite generating set $X$, there exist constants $\lambda,D > 1$ such that \[…

As well known, classical catenoids in ${\mathbf{R}}^3$ possess logarithmic growth at infinity. In this note we prove that the case of nonlocal minimal surfaces is significantly different, and indeed all nonlocal catenoids must grow at least…

We study the growth of typical groups from the family of $p$-groups of intermediate growth constructed by the second author. We find that, in the sense of category, a generic group exhibits oscillating growth with no universal upper bound.…

Let $G$ be a group that is relatively hyperbolic with respect to a collection of subgroups $\{H_{\lambda}\}_{\lambda\in \Lambda}$. Suppose that $G$ is given by a finite relative presentation $\mathcal{P}$ with respect to this collection. We…

Let G be a non-elementary torsion-free hyperbolic group. We prove that the exponential growth rate of the periodic quotient G/G^n tends to the one of G as n odd approaches infinity. Moreover we provide an estimate at which the convergence…

In this paper, we study the growth of confined subgroups through boundary actions of groups with contracting elements. We establish that the growth rate of a confined subgroup is strictly greater than half of the ambient growth rate in…

Given a group G acting on a geodesic metric space, we consider a preferred collection of paths of the space -- a path system -- and study the spectrum of relative exponential growth rates and quotient exponential growth rates of the…

Let $G$ be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let $N$ be an infinite normal subgroup of $G$, and let $\delta_N$ and $\delta_G$ be the growth rates of $N$ and $G$ with…

We show that every non-elementary group $G$ acting properly and cocompactly by isometries on a proper geodesic Gromov hyperbolic space $X$ is growth tight. In other words, the exponential growth rate of $G$ for the geometric…

We prove lower bounds on $||T_t||$, where $T_t$ is a one-parameter semigroup, starting from information on the resolvent norms, i.e. the pseudospectra. We provide a physically important example in which the growth of the semigroup norm…

We provide a lower bound for the uniform exponential growth rate of closed nonflat nonpositively curved 3-manifold groups. A detailed study of the uniform exponential growth rate of closed 3-manifold groups is also presented.

We give a subexponential upper bound and a superpolynomial lower bound on the growth function of the Fabrykowski-Gupta group. As a consequence, we answer negatively a question by Longobardi, Maj and Rhemtulla about characterizing groups…

In this paper we study the generic, i.e., typical, behavior of finitely generated subgroups of hyperbolic groups and also the generic behavior of the word problem for amenable groups. We show that a random set of elements of a nonelementary…

A group presentation is said to have rational growth if the generating series associated to its growth function represents a rational function. A long-standing open question asks whether the Heisenberg group has rational growth for all…

The topology of the Bowditch boundary of a relatively hyperbolic group pair gives information about relative splittings of the group. It is therefore interesting to ask if there is generic behavior of this boundary. The purpose of this…

More than thirty years ago, Brooks and Buser-Sarnak constructed sequences of closed hyperbolic surfaces with logarithmic systolic growth in the genus. Recently, Liu and Petri showed that such logarithmic systolic lower bound holds for every…

We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the…

Generalising results of Razborov and Safin, and answering a question of Button, we prove that for every hyperbolic group there exists a constant $\alpha >0$ such that for every finite subset $U$ that is not contained in a virtually cyclic…