Related papers: Fixed points of endomorphisms of graph groups

For every finitely generated free group $F$, we construct an irreducible open $3$-manifold $M_F$ whose end set is homeomorphic to a Cantor set, and with the end homogeneity group of $M_F$ isomorphic to $F$. The end homogeneity group is the…

The classical result by Dyer--Scott about fixed subgroups of finite order automorphisms of $F_n$ being free factors of $F_n$ is no longer true in $Z^m\times F_n$. Within this more general context, we prove a relaxed version in the spirit of…

We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacent matrix and automorphism group.

We prove that an automorphism $\phi:F\to F$ of a finitely generated free group $F$ is hyperbolic in the sense of Gromov if it has no nontrivial periodic conjugacy classes.

A group, $\fl{H}$, of automorphisms of a totally disconnected locally compact group, $G$, is flat if there is a compact open $U\leq G$ such that the index $[\alpha(U):U\cap \alpha(U)]$ is mininimized for every $\alpha\in\fl{H}$. The…

We study endomorphisms of a free group of finite rank by means of their action on specific sets of elements. In particular, we prove that every endomorphism of the free group of rank 2 which preserves an automorphic orbit (i.e., acts ``like…

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…

A finitely generated group $G$ is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety $\mathcal{UV}$, where $\mathcal{U}$ (respectively,…

We study when a group of form $G\times\mathbb{Z}^m (m\geq 1)$ has the finitely generated fixed subgroup property of automorphisms ($\rm{FGFP}_a$), by using the BNS-invariant, and provide some partial answers and non-trivial examples.

Given a countable group $G$ and two subshifts $X$ and $Y$ over $G$, a continuous, shift-commuting map $\phi : X \to Y$ is called a homomorphism. Our main result states that if every finitely generated subgroup of $G$ has polynomial growth,…

Let $A$ be an abelian variety over an algebraically closed field. We show that $A$ is the automorphism group scheme of some smooth projective variety if and only if $A$ has only finitely many automorphisms as an algebraic group. This…

Let $G$ be connected nilpotent Lie group acting locally on a real surface $M$. Let $\varphi$ be the local flow on $M$ induced by a $1$-parameter subgroup. Assume $K$ is a compact set of fixed points of $\varphi$ and $U$ is a neighborhood of…

Planar locally finite graphs which are almost vertex transitive are discussed. If the graph is 3-connected and has at most one end then the group of automorphisms is a planar discontinuous group and its structure is well-known. A general…

Let $F$ be a finitely generated free group. By using Bestvina-Handel theory, as well as some further improvements, the eigengroups of a given automorphism of $F$ (and its fixed subgroup among them) are globally analyzed and described. In…

Let $\phi:G\rightarrow G$ be an endomorphism of a finitely generated residually finite group. R.~Hirshon asked if there exists~$n$ such that the restriction of $\phi$ to $\phi^n(G)$ is injective. We give an example to show that this is not…

A group $G$ is said to satisfy the finitely generated intersection property (f.g.i.p.) if the intersection of any two finitely generated subgroups of $G$ is again finitely generated. The aim of this article is to understand when the…

We describe inertial endomorphisms of an abelian group $A$, that is endomorphisms $\varphi$ with the property $|(\varphi(X)+X)/X|<\infty$ for each $X\le A$. They form a ring containing multiplications, the so-called finitary endomorphisms…

An automorphism of a graph $G$ with $n$ vertices is a bijective map $\phi$ from $V(G)$ to itself such that $\phi(v_i)\phi(v_j)\in E(G)$ $\Leftrightarrow$ $v_i v_j\in E(G)$ for any two vertices $v_i$ and $v_j$ of $G$. Denote by…

Indices of fixed point classes play a central role in Nielsen fixed point theory. Jiang-Wang-Zhang proved that for selfmaps of graphs and surfaces, the index of any fixed point class has an upper bound called its characteristic. In this…

We give an explicit characterization of which direct products $G$ of surface groups of Euclidean type satisfy that the fixed subgroup of any automorphism (or endomorphism) of $G$ is compressed, and of which is it always inert.