Related papers: Generating the Mobius group with involution conjug…

Let X be a Hyperk\"{a}hler variety deformation equivalent to the Hilbert square on a K3 surface and let f be an involution preserving the symplectic form. We prove that the fixed locus of f consists of 28 isolated points and 1 K3 surface,…

Let the measure algebra of a topological group be equipped with the topology of uniform convergence on bounded right uniformly equicontinuous sets of functions. Convolution is separately continuous on the measure algebra, and it is jointly…

Inspired by results of Eskin and Mirzakhani counting closed geodesics of length $\le L$ in the moduli space of a fixed closed surface, we consider a similar question in the $Out(F_r)$ setting. The Eskin-Mirzakhani result can be equivalently…

Given a finite group with a generating subset there is a well-established notion of length for a group element given in terms of its minimal length expression as a product of elements from the generating set. Recently, certain quantities…

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices and let $\mathcal S$ be a generating set of $M_n(\mathbb{F})$ as an $\mathbb{F}$-algebra. The length of a finite generating set $\mathcal S$ of $M_n(\mathbb{F})$ is the smallest…

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 \[…

An element of a group is called $\textit{strongly reversible}$ or $\textit{strongly real}$ if it can be expressed as a product of two involutions. We provide necessary and sufficient conditions for an element of $\mathrm{SL}(n,\mathbb{C})$…

We study a class $\mathfrak{M}$ of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations. This class includes many finite metacyclic…

In [13], it is proved that any subgroup of $\mathrm{Diff}_{+}^{\omega }(I)$ (the group of orientation preserving analytic diffeomorphisms of the interval) is either metaabelian or does not satisfy a law. A stronger question is asked whether…

We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic $K$-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy…

A numerical semigroup is said to be ordinary if it has all its gaps in a row. Indeed, it contains zero and all integers from a given positive one. One can define a simple operation on a non-ordinary semigroup, which we call here the…

We study the question of existence of positive steady states of nonlinear evolution equations. We recast the steady state equation in the form of eigenvalue problems for a parametrised family of unbounded linear operators, which are…

Let $G$ be a group. Two elements $x, y$ are said to be {\it $z$-equivalent} if their centralizers are conjugate in $G$. The class equation of $G$ is the partition of $G$ into conjugacy classes. Further decomposition of conjugacy classes…

Let $G$ be a finite group. We consider the problem of counting simultaneous conjugacy classes of $n$-tuples and simultaneous conjugacy classes of commuting $n$-tuples in $G$. Let $\alpha_{G,n}$ denote the number of simultaneous conjugacy…

For each positive integer $k$ we present an example of Coxeter system $(G_k,S_k)$ such that $G_k$ is a word-hyperbolic Coxeter group, for any two generating reflections $s,t\in S_k$ the product $st$ has finite order, and the Coxeter graph…

Let $\Gamma_p$ denote the Hecke group where $p=2r$, $r>0$. Let $\mathcal{N}_l$ denote the set of conjugacy classes of reciprocal elements of word length $l$ in $\Gamma_p$. We prove that for $l \to \infty$, $$|\mathcal{N}_l| =…

Let $R$ be a semilocal principal ideal domain. Two algebraic objects over $R$ in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all…

We study the countable set of rates of growth of a hyperbolic group with respect to all its finite generating sets. We prove that the set is well-ordered, and that every real number can be the rate of growth of at most finitely many…

In this article we consider commuting graphs of involution conjugacy classes in the affine Weyl group of type $\tilde C_n$. We show that where the graph is connected the diameter is at most $n+2$.

Let G be a Lie group modelled on a locally convex space, with Lie algebra g, and k be a non-negative integer or infinity. We say that G is C^k-semiregular if each C^k-curve c in g admits a left evolution Evol(c) in G. If, moreover, the map…