Related papers: Linearly Bounded Conjugator Property for Mapping C…

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…

Let G be a connected reductive group over an algebraically closed field. We define a decomposition of G into finitely many strata such that each stratum is a union of conjugacy classes of fixed dimension; the strata are indexed by a set…

We study torsion generators for the (extended) mapping class group or the extended mapping class group of a closed connected orientable surface of genus g. We show that for every g is grater than or equal to 14, mapping class group can be…

Let $S$ be a semigroup. The elements $a,b\in S$ are called primarily conjugate if $a=xy$ and $b=yx$ for certain $x,y\in S$. The relation of conjugacy is defined as the transitive closure of the relation of primary conjugacy. In the case…

Let $G$ be a group. Two elements $x,y \in G$ are said to be in the same $z$-class if their centralizers in $G$ are conjugate within $G$. Consider $\mathbb F$ a perfect field of characteristic $\neq 2$, which has a non-trivial Galois…

We establish several new bounds for the number of conjugacy classes of a finite group, all of which involve the maximal number c of conjugacy classes of a normal subgroup fixed by some element of a suitable subset of the group. To apply…

Using combinatorial techniques, we answer two questions about simple classical Lie groups. Define $N(G,m)$ to be the number of conjugacy classes of elements of finite order $m$ in a Lie group $G$, and $N(G,m,s)$ to be the number of such…

Let $G$ be a classical group defined over a finite field. We consider the following fundamental problems concerning conjugacy in $G$: 1. List a representative for each conjugacy class of $G$. 2. Given $x \in G$, describe the centralizer of…

Let $H$ be an extension of a finite group $Q$ by a finite group $G$. Inspired by the results of duality theorems for \'etale gerbes on orbifolds, we describe the number of conjugacy classes of $H$ that maps to the same conjugacy class of…

Let G be a group. Two elements x and y in G are said to be in the same z-class if their centralizers in G are conjugate within G. In this paper, we prove that the number of z-classes in the group of upper triangular matrices is infinite…

A conjugacy class $C$ of a finite group $G$ is a sign conjugacy class if every irreducible character of $G$ takes value 0, 1 or -1 on $C$. In this paper we classify the sign conjugacy classes of alternating groups.

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…

The mapping class group of a compact oriented surface of genus greater than one with boundary acts ergodically on connected components of the representation variety corresponding to a connected compact Lie group, for every choice of…

The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves…

We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that…

An element w in the free group on r letters defines a map f from G^r to G for each group G. In this note, we show that whenever w is non-trivial and G is a semisimple algebraic group, f is dominant. When G is a finite simple group, the…

The excess of an element $w$ of a finite Coxeter group $W$ is the minimal value of $l(x) + l(y) - l(w)$, where $x$, $y$ are elements of $W$ such that $x^2 = y^2 = 1$ and $w = xy$. Every element of a finite Coxeter group is either an…

Let G be an affine algebraic group over an algebraically closed field such that the identity component G^0 of G is reductive. Let W be the Weyl group of G and let D be a connected component of G whose image in G/G^0 is a unipotent element.…

We study the representation growth of alternating and symmetric groups in positive characteristic and restricted representation growth for the finite groups of Lie type. We show that the the number of representations of dimension at most n…

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a ``matrix problem''. Such problems involve finding normal forms for matrices under a specified set of row and…