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

We prove the congruence subgroup property for the centralizer of a finite subgroup $G$ in the mapping class group of a hyperbolic oriented and connected surface of finite topological type $S$ such that the genus of the quotient surface…

Let $\Gamma$ be a finite index subgroup of the mapping class group $MCG(\Sigma)$ of a closed orientable surface $\Sigma$, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element…

In this article we show that for any given Riemann surface $\Sigma$ of genus $g$, we can bound (from above) the renormalized volume of a (hyperbolic) Schottky group with boundary at infinity conformal to $\Sigma$ in terms of the genus and…

For any positive integer $g$, we completely determine the minimal genus function for $\Sigma_{g}\times T^{2}$. We show that the lower bound given by the adjunction inequality is not sharp for some class in $H_{2}(\Sigma_{g}\times T^{2})$.…

Any nontrivial homomorphism from the mapping class group of an orientable surface of genus $g\geq 3$ to $\GL(2g,\C)$ is conjugate to the standard symplectic representation. It is also shown that the mapping class group has no faithful…

Let $\Sigma_{g,n}$ be a compact oriented surface with genus $g\geq 2$ bordered by $n$ circles. Due to Witten, the twisted Reidemeister torsion coincides with a power of the Atiyah-Bott-Goldman-Narasimhan symplectic form on the space of…

We study widths of conjugacy classes in anisotropic higher rank $S$-arithmetic groups of orthogonal type. Assuming the GRH, we prove that many such groups have bounded conjugacy width. For example, this holds if the degree is greater or…

Let $G$ be the first Grigorchuk group. We show that the commutator width of $G$ is $2$: every element $g\in [G,G]$ is a product of two commutators, and also of six conjugates of $a$. Furthermore, we show that every finitely generated…

We construct the first examples of finitely presented groups where the conjugator length function is exponential; these are central extensions of groups of the form $F_m \rtimes F_2$. Further, we use a fibre product construction to exhibit…

Let $\Sigma_g$ be a compact, connected, orientable surface of genus $g \geq 2$. We ask for a parametrization of the discrete, faithful, totally loxodromic representations in the deformation space ${\rm Hom}(\pi_1(\Sigma_g), {\rm…

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group $\Gamma$ of a closed hyperbolic surface $\Sigma$ in ${\rm PSL}(2,\mathbb R)^n$. We identify the boundary…

Let $S$ be a minimal surface of general type with irregularity $q(S) = 1$. Well-known inequalities between characteristic numbers imply that $3 p_g(S) \le c_2(S) \le 10 p_g(S)$, where $p_g(S)$ is the geometric genus and $c_2(S)$ the…

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces $f:\Sigma \to \R^3$ are nondegenerate; that is, the Jacobi operator $\Delta_f + |A_f|^2$ has no $L^2$ kernel. In fact, if $\Sigma$ has genus zero and…

Given a hyperelliptic hyperbolic surface $S$ of genus $g \geq 2$, we find bounds on the lengths of homologically independent loops on $S$. As a consequence, we show that for any $\lambda \in (0,1)$ there exists a constant $N(\lambda)$ such…

Let $g, n \geq 0$ and $\Sigma = \Sigma_{g, n}$ be a connected oriented surface of genus $g$ with $n$ punctures. The $\mathrm{SL}_2$-character variety of $\Sigma$ has a rigid relative automorphism group, whose elements fix each monodromies…

For $r \in \mathbb{R}, r> 1$ and $n \in \mathbb{Z}^+$, the divisor function $\sigma_{-r}$ is defined by $\sigma_{-r}(n) := \sum_{d \vert n} d^{-r}$. In this paper we show the number $C_r$ of connected components of…

Every finite group whose order is divisible by a prime $p$ has at least $2 \sqrt{p-1}$ conjugacy classes.

In any Coxeter group, the conjugates of elements in the standard minimal generating set are called reflections and the minimal number of reflections needed to factor a particular element is called its reflection length. In this article we…

Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the…

We classify representations of the mapping class group of a surface of genus $g$ (with at most one puncture or boundary component) up to dimension $3g-3$. Any such representation is the direct sum of a representation in dimension $2g$ or…