Related papers: An Explicit Bound for the Conjugator Length Functi…

In this paper, we primarily investigate the following symmetric presentation of the surface group $\pi_1(\Sigma_g)=\left\langle c_1,\dots, c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\right\rangle$. For every nontrivial element…

The conjugator length function of a finitely generated group is the function $f$ so that $f(n)$ is the minimal upper bound on the length of a word realizing the conjugacy of two words of length at most $n$. We study herein the spectrum of…

We describe the topological behavior of the conjugacy action of the mapping class group of an orientable infinite-type surface $\Sigma$ on itself. Our main results are: (1) All conjugacy classes of $MCG(\Sigma)$ are meager for every…

Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study the behaviour of conjugacy length…

We show that the conjugator length function of the Baumslag-Gersten group is bounded above and below by a tower of exponentials of logarithmic height -- in particular it grows faster than any tower of exponentials of fixed height. We…

Given a hyperbolic surface $\Sigma$ of genus $g$ with $r$ cusps, Mirzakhani proved that the number of closed geodesics of length at most $L$ and of a given type is asymptotic to $cL^{6g-6+2r}$ for some $c>0$. Since a closed geodesic…

The conjugacy growth function counts the number of distinct conjugacy classes in a ball of radius $n$. We give a lower bound for the conjugacy growth of certain branch groups, among them the Grigorchuk group. This bound is a function of…

Modelled on efficient algorithms for solving the conjugacy problem in hyperbolic groups, we define and study the permutation conjugacy length function. This function estimates the length of a short conjugator between words $u$ and $v$, up…

Recently, Korkmaz established the lower bound of $3g - 2$ for the dimension of a faithful representation of the mapping class group of an orientable surface of genus $g \ge 3$. We raise this bound to $4g - 3$ in the setting of surfaces of…

The conjugator length function of a finitely generated group $\Gamma$ gives the optimal upper bound on the length of a shortest conjugator for any pair of conjugate elements in the ball of radius $n$ in the Cayley graph of $\Gamma$. We…

Let $g\geq3$ and $n\geq0$, and let ${\mathcal{M}}_{g,n}$ be the mapping class group of a surface of genus $g$ with $n$ boundary components. We prove that ${\mathcal{M}}_{g,n}$ contains a unique subgroup of index $2^{g-1}(2^{g}-1)$ up to…

We study conjugacy limits of certain of subgroups inside $\SL(2,\R)\ltimes\R^2$. These subgroups have a common feature that any two in the same category are conjugates of each other.

For a non-cyclic finite group $G$, let $\gamma(G)$ denote the smallest number of conjugacy classes of proper subgroups of $G$ needed to cover $G$. Bubboloni, Praeger and Spiga, motivated by questions in number theory, have recently…

We consider locally compact subgroups $H$ of the full isometry group $\mathrm{Isom}(\mathbb{E}^n)$ of Euclidean $n$-space which respect the splitting into an orthogonal and a translation subgroup. We prove that the conjugator length…

We prove that the number of distinct group actions on compact Riemann surfaces of a fixed genus $\sigma \geq 2$ is at least quadratic in $\sigma$. We do this through the introduction of a coarse signature space, the space…

We prove that a Kleinian surface groups is determined, up to conjugacy in the isometry group of $\mathbb H^3$, by its simple marked length spectrum. As a first application, we show that a discrete faithful representation of the fundamental…

This paper has two parts, on Baumslag-Solitar groups and on general G-trees. In the first part we establish bounds for stable commutator length (scl) in Baumslag-Solitar groups. For a certain class of elements, we further show that scl is…

The mapping class group $\mathrm{Mod}_{g, 1}$ of a surface with one marked point can be identified with an index two subgroup of $\mathrm{Aut}(\pi_1 \Sigma_g)$. For a surface of genus $g \geq 2$, we show that any action of $\mathrm{Mod}_{g,…

A limit group is the limit of a sequence of conjugates of the diagonal Cartan subgroup, C, of SL(3,R). We show C has 5 possible limit groups, up to conjugacy. Each limit group is determined by an equivalence class of nonstandard triangle,…

We provide a new upper bound on the number of conjugacy classes in the group $U_n(q)$ of unitriangular matrices over a finite field. We also compute a similar upper bound for every group in the lower central series of $U_n(q)$.