Related papers: Folding free-group automorphisms

A word in a group is called a test element if any endomorphism fixing it is necessarily an automorphism. In this note, we give a sufficient condition in geometry to construct test elements for monomorphisms of a free group, by using the…

We generalize the positive solution of the Frobenius conjecture and refinements thereof by studying the structure of groups that admit a fix-point-free automorphism satisfying an identity. We show, in particular, that for every polynomial…

Part 1 : We remark that the conjugacy problem for pairs of hyperbolic au- tomorphisms of a finitely presented group (typically a free group) is decidable. The solution that we propose uses the isomorphism problem for the suspensions, and…

By a map we mean a $2$-cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. Automorphism of a map can be thought of as a permutation of the vertices which…

A countable graph is ultrahomogeneous if every isomorphism between finite induced subgraphs can be extended to an automorphism. Woodrow and Lachlan showed that there are essentially four types of such countably infinite graphs: the random…

Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with…

We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be…

We study the automorphisms of graph products of cyclic groups, a class of groups that includes all right-angled Coxeter and right-angled Artin groups. We show that the group of automorphism generated by partial conjugations is itself a…

In this paper, we are not limited to considering semi-scalar automorphisms of G as Lindsay N. Childs in 2012 which is a specific case of Aut(G). Some new results in determining the fixed point free automorphisms of G are developed.

Let $G$ be an infinite simple group of finite Morley rank and $\alpha$ a supertight automorphism of $G$ so that the fixed point subgroup $P_n:=C_G(\alpha^n)$ is pseudofinite for all $n\in \mathbb{N}\setminus\{0\}$. It is know (using CFSG)…

The automorphism groups ${\rm Aut\,}A_n$ and ${\rm Aut\,}W_n$ of the polynomial algebra $A_n=C[x_1,x_2,\cdots, x_n]$ and the rank $n$ Witt algebra $W_n={\rm Der\,}A_n$ are studied in this paper. It is well-known that ${\rm Aut\,}A_n$ for…

This paper uses the methods from arXiv:0804.1396 to give 'short' presentations for Aut$^+(F_n)$, the special automorphism group of the free group of rank $n$, and Out $^+(F_n)$ the special outer automorphism group of the free group of rank…

We adapt the Ping-Pong Lemma, which historically was used to study free products of groups, to the setting of the homeomorphism group of the unit interval. As a consequence, we isolate a large class of generating sets for subgroups of…

In this paper, we explore the behaviour of the fixed subgroups of endomorphisms of free-abelian times free (FATF) groups. We exhibit an algorithm which, given a finitely generated subgroup $\mathcal{H}$ of a FATF group $\mathcal{G}$,…

We find a set of generators for the automorphism group of a graph product of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup of star-automorphisms defined…

An automorphism $\alpha$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively…

Let $(X, \sigma)$ be a transitive sofic shift and let $\rm{Aut}(X)$ denote its automorphism group. We generalize a result of Frisch, Schlank, and Tamuz to show that any normal amenable subgroup of $\rm{Aut}(X)$ must be contained in the…

We classify abelian subgroups of Out(F_n) up to finite index in an algorithmic and computationally friendly way. A process called disintegration is used to canonically decompose a single rotationless element \phi into a composition of…

We provide an effective algorithm for determining whether an element of the outer automorphism group of a free group is fully irreducible. Our method produces a finite list which can be checked for periodic proper free factors.

We prove Nielsen realisation for finite subgroups of the groups of untwisted outer automorphisms of RAAGs in the following sense: given any graph $\Gamma$, and any finite group $G\leqslant \mathrm{U}^0(A_\Gamma) \leqslant…