Related papers: Recognition and constructive membership for purely…

In the 1980's Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse…

If $G$ is a free product of finite groups, let $\Sigma Aut_1(G)$ denote all (necessarily symmetric) automorphisms of $G$ that do not permute factors in the free product. We show that a McCullough-Miller [D. McCullough and A. Miller, {\em…

We study in detail the profinite group G arising as geometric \'etale iterated monodromy group of an arbitrary quadratic polynomial over a field of characteristic different from two. This is a self-similar closed subgroup of the group of…

Consider a smooth connected algebraic group $G$ acting on a normal projective variety $X$ with an open dense orbit. We show that Aut($X$) is a linear algebraic group if so is $G$; for an arbitrary $G$, the group of components of Aut($X$) is…

For a hyperbolic toral automorphism, we construct a profinite completion of an isomorphic copy of the homoclinic group of its right action using isomorphic copies of the periodic data of its left action. The resulting profinite group has a…

In this paper we survey recent developments in the theory of groups acting on $\Lambda$-trees. We are trying to unify all significant methods and techniques, both classical and recently developed, in an attempt to present various faces of…

We prove that many dynamical properties of group cellular automata (i.e., cellular automata defined on any finite group and with global rule which is an endomorphism), including surjectivity, injectivity, sensitivity to initial conditions,…

We give a combinatorial criterion that implies both the non-strong relative hyperbolicity and the one-endedness of a finitely generated group. We use this to show that many important classes of groups do not admit a strong relatively…

We give an algorithm for finding the index of a positive outer automorphism of the free group, and prove the algorithm exits in a finite time.

In the present paper we continue studying regular free group actions on $\mathbb{Z}^n$-trees. We show that every finitely generated $\mathbb{Z}^n$-free group $G$ can be embedded into a finitely generated $\mathbb{Z}^n$-free group $H$ acting…

A group $G$ is said to be a {\it CSA}-group if all maximal abelian subgroups of $G$ are malnormal. The class of CSA groups is of interest because it contains torsion-free hyperbolic groups, groups acting freely on $\Lambda$-trees and groups…

The following discourse is inspired by the works on hyperbolic groups of Epstein, and Neumann/Reeves. Epstein showed that geometrically finite hyperbolic groups are biautomatic. Neumann/Reeves showed that virtually central extensions of…

We study the positive theory of groups acting on trees and show that under the presence of weak small cancellation elements, the positive theory of the group is trivial, i.e. coincides with the positive theory of a non-abelian free group.…

In this paper we discuss several heuristic strategies which allow one to solve the Whitehead's minimization problem much faster (on most inputs) than the classical Whitehead algorithm. The mere fact that these strategies work in practice…

The isotrivial Mordell-Lang theorem of Moosa and Scanlon describes the set $X\cap\Gamma$ when $X$ is a subvariety of a semiabelian variety $G$ over a finite field $\mathbb{F}_q$ and $\Gamma$ is a finitely generated subgroup of $G$ that is…

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…

We show that the generation problem in Thompson group $F$ is decidable, i.e., there is an algorithm which decides if a finite set of elements of $F$ generates the whole $F$. The algorithm makes use of the Stallings $2$-core of subgroups of…

As defined by Guralnick and Saxl given a nonabelian simple group $S$ and its nonidentity automorphism $x$, a natural number $\alpha_S(x)$ does not exceed a natural number $m$ if some $m$ conjugates of $x$ in the group $\langle x,S\rangle$…

In evolutionary biology, biologists often face the problem of constructing a phylogenetic tree on a set $X$ of species from a multiset $\Pi$ of partitions corresponding to various attributes of these species. One approach that is used to…

Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring K[X]^G in the case where G is reductive. Furthermore, we address the case where G is connected and…