Related papers: A generating set for the palindromic Torelli group

We prove that a semi-direct product of two finite rank free groups $F_k$ and $F_n$ such that $F_k$ acts on $F_n$ by polynomially growing automorphisms acts properly isometrically on a finite dimensional CAT(0) cube complex provided some…

The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class…

We study the automorphism groups of free-by-cyclic groups and show these are finitely generated in the following cases: (i) when defining automorphism has linear growth and (ii) when the rank of the underlying free group has rank at most 3.…

Let $\Lambda_0$ be an ordered abelian group. We show how an $\mathrm{ATF}(\mathbb{Z}\times\Lambda_0)$ group -- that is, a group admitting a free affine action without inversions on a $\mathbb{Z}\times\Lambda_0$-tree -- admits a natural…

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^{1+epsilon} where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple…

Let $\FP(X)$ be the free paratopological group on a topological space $X$. For $n\in \N$, denote by $\FP_n(X)$ the subset of $\FP(X)$ consisting of all words of reduced length at most $n$, and by $i_n$ the natural mapping from $(X\oplus…

Let $N$ be a finitely generated nilpotent group. Algorithm is constructed such, that for every automorphism $\phi \in Aut(N)$ defines the Reidemeister number $R(\phi).$ It is proved that any free nilpotent group of rank $r = 2$ or $r = 3$…

Let ${\cal M}_{g,n}$ and ${\cal H}_{g,n}$, for $2g-2+n>0$, be, respectively, the moduli stack of $n$-pointed, genus $g$ smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be…

If $F$ is a finitely generated free group and $\phi$ is a positive automorphism of $F$ then $F\rtimes_\phi Z$ satisfies a quadratic isoperimetric inequality.

In this paper, the problem formulated in [8] is solved. We prove, that the group of automorphisms of the category of free associative algebras is generated by semi-inner and mirror automorphisms

Let $FP(X)$ be the free paratopological group over a topological space $X$. For each non-negative integer $n\in\mathbb{N}$, denote by $FP_{n}(X)$ the subset of $FP(X)$ consisting of all words of reduced length at most $n$, and $i_{n}$ by…

To any free group automorphism, we associate a universal (cone of) limit tree(s) with three defining properties: first, the tree has a minimal isometric action of the free group with trivial arc stabilizers; second, there is a unique…

For any surface $\Sigma$ of infinite topological type, we study the Torelli subgroup ${\mathcal I}(\Sigma)$ of the mapping class group ${\rm MCG}(\Sigma)$, whose elements are those mapping classes that act trivially on the homology of…

The twin group $T_n$ is a Coxeter group generated by $n-1$ involutions and the pure twin group $PT_n$ is the kernel of the natural surjection of $T_n$ onto the symmetric group on $n$ letters. In this paper, we investigate structural aspects…

The observation that a graph of rank $n$ can be assembled from graphs of smaller rank $k$ with $s$ leaves by pairing the leaves together leads to a process for assembling homology classes for $Out(F_n)$ and $Aut(F_n)$ from classes for…

Fixed points ${\bf u}=\varphi({\bf u})$ of marked and primitive morphisms $\varphi$ over arbitrary alphabet are considered. We show that if ${\bf u}$ is palindromic, i.e., its language contains infinitely many palindromes, then some power…

We study the diagram alphabet of knot moves associated with the character rings of certain matrix groups. The primary object is the Hopf algebra Char-GL of characters of the finite dimensional polynomial representations of the complex group…

We show how to construct a family of groups with simple commutator subgroups from aperiodic 1-vertex, finitely aligned higher rank graphs (which are, in fact, a class of cancellative monoids). Inverse semigroups form the intermediary…

We compute the two-cocycles (or multipliers) of the free nilpotent groups of class $2$ and rank $n$ and give conditions for simplicity of the corresponding twisted group $C^*$-algebras. These groups are representation groups for…

We study sets of solutions to equations over a free group, projections of such sets, and the structure of elementary sets defined over a free group. The structre theory we obtain enable us to answer some questions of A. Tarski's, and…