Related papers: Finitely generated infinite simple groups of infin…

Say that a finite group $G$ is mixable if a product of random elements, each chosen independently from two options, can distribute uniformly on $G$. We present conditions and obstructions to mixability. We show that $2$-groups, the…

We combine classical methods of combinatorial group theory with the theory of small cancellations over relatively hyperbolic groups to construct finitely generated torsion-free groups that have only finitely many classes of conjugate…

We provide new bounds for the divisibility function of the free group F_2 and construct short laws for the symmetric groups Sym(n). The construction is random and relies on the classification of the finite simple groups. We also give bounds…

The conjugator length function of a finitely generated group is the function $f$ so that $f(n)$ is the minimal upper bound on the length of a word realizing the conjugacy of two words of length at most $n$. We study herein the spectrum of…

We prove, for various important classes of Mealy automata, that almost all generated groups have an element of infinite order. In certain cases, it also implies other results such as exponential growth.

We exhibit finitely generated torsion-free groups for which any action on any finite-dimensional CW-complex with finite Betti numbers has a global fixed point.

Big mapping class groups are the mapping class groups of infinite-type surfaces, that is, surfaces whose fundamental groups are not finitely generated. While mapping class groups of finite-type surfaces have been extensively studied, the…

We prove that for a suitable class of representations of free group tensor products are generically irreducible. In particular we prove that there exist irreducible boundary realizations with infinite dimensional fiber.

We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely generated residually finite group. The same holds for Cartesian producs of other simple groups…

We investigate the isolated points in the space of finitely generated groups. We give a workable characterization of isolated groups and study their hereditary properties. Various examples of groups are shown to yield isolated groups. We…

We examine the relationship between finitely and infinitely generated relatively hyperbolic groups, in two different contexts. First, we elaborate on a remark from math.GR/0601311, which states that the version of Dehn filling in relatively…

We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of…

A recent conjecture of the author and Teng Fang states that there are only finitely many finite simple groups with no cubic graphical regular representation. In this paper, we make a crucial progress towards this conjecture by giving an…

We prove that the Fesenko Groups, which occur as natural closed subgroups of the Nottingham Group, have finite width and infinite obliquity.

Necessary and sufficient conditions for finite semihypergroups to be built from groups of the same order are established

Let $\mathcal G$ denote the space of finitely generated marked groups. For any finitely generated group $G$, we construct a continuous, injective map $f$ from the space of subgroups $Sub(G)$ to $\mathcal G$ that sends conjugate subgroups to…

Given an arbitrary group $G$ we construct a semigroup of idempotents (band) $B_G$ with the property that the free idempotent generated semigroup over $B_G$ has a maximal subgroup isomorphic to $G$. If $G$ is finitely presented then $B_G$ is…

Let G be a finite group. It has recently been proved that every nontrivial element of G is contained in a generating set of minimal size if and only if all proper quotients of G require fewer generators than G. It is natural to ask which…

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over $\Z$ cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers of an infinite order…

A B-group is a group such that all its minimal generating sets (with respect to inclusion) have the same size. We prove that the class of finite B-groups is closed under taking quotients and that every finite B-group is solvable. Via a…