Related papers: On hopfian cofinite subsemigroups

Every countable group $G$ can be embedded in a finitely generated group $G^*$ that is hopfian and complete, i.e. $G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is…

A group $\Gamma$ is defined to be cofinitely Hopfian if every homomorphism $\Gamma\to\Gamma$ whose image is of finite index is an automorphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic…

A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups G that admit a descending chain of proper normal finite-index subgroups, each of which is…

We call a semigroup $S$ f-noetherian if every right congruence of finite index on $S$ is finitely generated. We prove that every finitely generated semigroup is f-noetherian, and investigate whether the properties of being f-noetherian and…

We obtain some general restrictions on the continuous endomorphisms of a profinite group G under the assumption that G has only finitely many open subgroups of each index (an assumption which automatically holds, for instance, if G is…

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. We say that $H$ is s-semipermutable in $G$ if $HG_p = G_pH$ for any Sylow $p$-subgroup $G_p$ of $G$ with $(p, |H|) = 1$. We investigate the influence of s-semipermutable…

Let G be a finitely generated relatively hyperbolic group. We show that if no peripheral subgroup of G is hyperbolic relative to a collection of proper subgroups, then the fixed subgroup of every automorphism of G is relatively quasiconvex.…

We prove that every finitely generated soluble group which is not virtually abelian has a subgroup of one of a small number of types.

We find a non-Hopfian ascending HNN-extension of a finitely presented Hopfian group by providing an explicit construction. This result addresses an analogous question to the one posed by Sapir and Wise, which asks whether there is a…

If a finitely generated torsion free group K has the property that all finitely generated subgroups S of K are either small or have growth constant bounded uniformly away from 1 then a non proper HNN extension G of K, that is a semidirect…

We exhibit two finitely generated residually finite groups $G$ and $H$ with isomorphic profinite completions $\widehat{G} \cong \widehat{H}$, such that $G$ is co-Hopfian while $H$ is not. The construction utilizes Wise's residually finite…

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. (1) If G is a finitely generated non-elementary relatively hyperbolic group…

Given an inverse semigroup $G_0$ of bounded type, we show, along with some other assumptions, that if the set of incompressible elements of $G_0$ is finite, then any finitely generated subgroup $G$ of the topological full group…

The main purpose of the present work is an investigation of the notions Hopfian (co-Hopfian) acts whose their surjective (injective) endomorphisms are isomorphisms. While we investigate conditions that are relevant to these classes of acts,…

A self-similar group of finite type is the profinite group of all automorphisms of a regular rooted tree that locally around every vertex act as elements of a given finite group of allowed actions. We provide criteria for determining when a…

For each of the following conditions, we characterize the pseudovarieties of semigroups V that satisfy it: (i) every epimorphism to a member of V is onto; (ii) every epimorphism to a finite semigroup with domain a member of V is onto; (iii)…

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…

Let S be a connected, compact and orientable surface of genus two having exactly one boundary component. We study automorphisms of the Torelli complex for S, and describe any isomorphism between finite index subgroups of the Torelli group…

Let $S$ be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of $S$ is finite and at least 4, then the isomorphism type of the pure mapping class group associated to $S$,…

We produce an example demonstrating that every finitely generated relatively hyperbolic group with respect to a collection of Hopfian subgroups need not be Hopfian. This answers a question of Osin \cite[Problem 5.5]{Osin} in the negative.