Related papers: Statistics in conjugacy classes in free groups

We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only on vector spaces. Some examples and…

The purpose of this paper is to investigate torsion-free groups which act properly and cocompactly on CAT(0) metric spaces which have isolated flats, as defined by Hruska. Our approach is to seek results analogous to those of Sela,…

In this technical note we describe a pair of results on heterotic compactifications. First, we give an example demonstrating that the usual statement of the anomaly-freedom constraint for perturbative heterotic compactifications (meaning,…

We prove a Tits alternative theorem for groups acting on CAT(0) cubical complexes. Namely, suppose that $G$ is a group for which there is a bound on the orders of its finite subgroups. We prove that if $G$ acts properly on a…

We establish the rationality of the stable conjugation-invariant word norm on free groups and virtually free Coxeter groups.

We prove a central limit theorem for a sequence of random variables whose means are ambiguous and vary in an unstructured way. Their joint distribution is described by a set of measures. The limit is (not the normal distribution and is)…

The quotients $G_k/G_{k+1}$ of the lower central series of a finitely presented group $G$ are an important invariant of this group. In this work we investigate the ranks of these quotients in the case of a certain class of conjugation-free…

To every finitely generated group one can assign the conjugacy growth function that counts the number of conjugacy classes intersecting a ball of radius $n$. Results of Ivanov and Osin show that the conjugacy growth function may be constant…

The exploration of associations between random objects with complex geometric structures has catalyzed the development of various novel statistical tests encompassing distance-based and kernel-based statistics. These methods have various…

Using the concept of algebraically closed groups, we prove that there is a countable torsion free group with exactly two conjugacy classes.

Let H and K be quasiconvex subgroups of a negatively curved torsion-free group G. We give an algorithm which decides whether an element of H is conjugated in G to an element of K.

Oriented loops on an orientable surface are, up to equivalence by free homotopy, in one-to-one correspondence with the conjugacy classes of the surface's fundamental group. These conjugacy classes can be expressed (not uniquely in the case…

In this paper, we consider stochastic dual coordinate (SDCA) {\em without} strongly convex assumption or convex assumption. We show that SDCA converges linearly under mild conditions termed restricted strong convexity. This covers a wide…

Given a conjugacy class $\mathcal{C}$ in a group $G$ we define a new graph, $\Gamma(\mathcal{C})$, whose vertices are elements of $\mathcal{C}$; two vertices $g,h\in \mathcal{C}$ are connected in $\Gamma(\mathcal{C})$ if $[g,h]=1$ and…

In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is $0$ for all…

In this paper, we consider the conjugacy growth function of a group, which counts the number of conjugacy classes which intersect a ball of radius $n$ centered at the identity. We prove that in the case of virtually polycyclic groups, this…

In this paper, we examine Lie group actions on moduli spaces (sets themselves built as quotients by group actions) and their fixed points. We show that when the Lie group is compact and connected, we obtain a linear constraint. This…

We introduce a notion of "freely braided element" for simply laced Coxeter groups. We show that an arbitrary group element $w$ has at most $2^{N(w)}$ commutation classes of reduced expressions, where $N(w)$ is a certain statistic defined in…

We discuss CLT for the global and local linear statistics of random matrices from classical compact groups. The main part of our proofs are certain combinatorial identities much in the spirit of works by Kac and Spohn.

Given a group $G$ and a subset $X \subset G$, an element $g \in G$ is called quasi-positive if it is equal to a product of conjugates of elements in the semigroup generated by $X$. This notion is important in the context of braid groups,…