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We extend Gow's theorem on products of semisimple regular conjugacy classes to finite groups whose generalized Fitting subgroup is Z(G)S where S is a quasisimple group of Lie type in characteristic p and Z(G) has order prime to p.

In this paper we provide an alternative solution to a result by Juh\'{a}sz that the twisted conjugacy problem for odd dihedral Artin groups is solvable, that is, groups with presentation $G(m) = \langle a,b \; | \; _{m}(a,b) = {}_{m}(b,a)…

A group $G$ has $FW_n$ if every action on a $n$-dimensional $\mathrm{CAT}(0)$ cube complex has a global fixed point. This provides a natural stratification between Serre's $FA$ and Kazhdan's $(T)$. For every $n$, we show that random groups…

Let $\mathcal{C}$ be a conjugacy class of involutions in a group $G$. We study the graph $\Gamma(\mathcal{C})$ whose vertices are elements of $\mathcal{C}$ with $g,h\in\mathcal{C}$ connected by an edge if and only if $gh\in\mathcal{C}$. For…

A group $G$ is called residually finite if for every non-trivial element $g \in G$, there exists a finite quotient $Q$ of $G$ such that the element $g$ is non-trivial in the quotient as well. Instead of just investigating whether a group…

We study quotients of mapping class groups of punctured spheres by suitable large powers of Dehn twists, showing an analogue of Ivanov's theorem for the automorphisms of the corresponding quotients of curve graphs. Then we use this result…

We prove that the iterated monodromy group of the polynomial $z^2+i$ is just-infinite, regular branch and does not have the congruence subgroup property. This yields the first example of an iterated monodromy group of a polynomial with…

We define the Cayley graph and its growth function for multivalued groups. We prove that if we change a finite set of generators of multivalued group, or change the starting point, we get an equivalent growth function. We prove that if we…

A well-known result of Shalom says that lattices in SO$(n,1)$ are $\mathrm{L}^p$ measure equivalent for all $p<n-1$. His proof actually yields the following stronger statement: the natural coupling resulting from a suitable choice of…

We show that the twisted conjugacy problem is solvable for large-type Artin groups whose outer automorphism group is finite, generated by graph automorphisms and the global inversion. This includes XXXL Artin groups whose defining graph is…

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| =…

We define the twofold semidirect product of two skew left braces, in which both the additive and multiplicative groups are semidirect products of the corresponding groups of the given skew left braces. We consider an odd prime $p$ and an…

Let $\mathbf{F}$ be the free group on two generators $a, b$ and let a family of words $w = [[a, b], [a^3, b^n]]$ in $\mathbf{F}$. In this paper we examine surjectivity of word map $w$ on special unitary group SU(2) over complex field…

In this paper we treat faithful actions of simple algebraic groups on irreducible modules and on the associated Grassmannian varieties. By explicit calculation, we show that in each case, with essentially one exception (only in…

We classify compactly generated locally compact groups of polynomial growth up to $L^p$ measure equivalence (ME) for all $p\leq 1$. To achieve this, we combine rigidity results (previously proved for discrete groups by Bowen and Austin)…

This paper investigates $2$-$(v,5,\lambda)$ designs $\mathcal{D}$ admitting a block-transitive automorphism group $G$. We first prove that if $G$ is point-imprimitive, then $v$ must be one of 16, 21, or 81. We further provide a complete…

We compare four different types of realizability for saturated fusion systems over discrete $p$-toral groups. For example, when $G$ is a locally finite group all of whose $p$-subgroups are artinian (hence discrete $p$-toral), we show that…

Given a profinite group $G$ and a family $\mathcal{F}$ of finite groups closed under taking subgroups, direct products and quotients, denote by $\mathcal{F}(G)$ the set of elements $g \in G$ such that $\{x \in G\ |\ \langle g,x \rangle \…

We establish that finitely generated non-abelian direct products $G$ of free pro-$p$ groups have full Hausdorff spectrum with respect to the lower $p$-series $\mathcal{L}$. This complements similar results with respect to other standard…

In 1976, L.N. Vaserstein used a construction analogous to the Gram-Schmidt orthogonalisation, for obtaining a set of symplectic matrices from a set of elementary matrices. We have a similar construction for Petrov's odd unitary group. Here,…