We describe the endomorphisms of the direct product of two free groups of finite rank and obtain conditions for which the subgroup of fixed points is finitely generated and we do the same for periodic points. We also describe the automorphisms of such a group and solve the three Whitehead problems for automorphisms, monomorphisms and endomorphisms for groups in this class. We also obtain conditions for an endomorphism to be uniformly continuous when a suitable metric is considered and study infinite fixed points of the extension to the completion of the group.
Let $G=\ast_{i=1}^{n}G_{i}$ and let $ϕ$ be a symmetric endomorphism of $G$. If $ϕ$ is a monomorphism or if $G$ is a finitely generated residually finite group, then the fixed subgroup $Fix(ϕ)=\{g\in G:ϕ(g)=g\}$ of $ϕ$ has Kurosh rank at most $n$.
We investigate the endomorphism monoids of certain finite $p$-groups of order $p^8$ first studied by Jonah and Konvisser in 1975 as examples for finite $p$-groups with abelian automorphism group, and we show some necessary conditions for a finite $p$-group to have commutative endomorphism monoid. As a by-product, apart from formulas for the number of conjugacy classes of endomorphisms of said groups, we will be able to derive the following: There exist nonabelian groups with commutative endomorphism monoid, and having commutative endomorphism monoid is a group property strictly stronger than h…
It is shown, for a given graph group $G$, that the fixed point subgroup Fix$\,φ$ is finitely generated for every endomorphism $φ$ of $G$ if and only if $G$ is a free product of free abelian groups. The same conditions hold for the subgroup of periodic points. Similar results are obtained for automorphisms, if the dependence graph of $G$ is a transitive forest.
Determinant for automorphisms of Zappa-Szép product of groups
Desh Bhagat Pandit Chetan Dev Government College of Education, Faridkot, Punjab, India（法里德科特德斯·巴哈格特·潘迪特·切坦·德夫政府教育学院）；Manipal University, Jaipur, Rajasthan, India（斋浦尔马匹阿尔大学）；Central University of Rajasthan, Ajmer, Rajasthan, India（阿杰梅尔拉贾斯坦中央大学）
Aiming at a better understanding of finite groups as finite dynamical systems, we show that by a version of Fitting's Lemma for groups, each state space of an endomorphism of a finite group is a graph tensor product of a finite directed $1$-tree whose cycle is a loop with a disjoint union of cycles, generalizing results of Hernández-Toledo on linear finite dynamical systems, and we fully characterize the possible forms of state spaces of nilpotent endomorphisms via their "ramification behavior". Finally, as an application, we will count the isomorphism types of state spaces of endomorphisms of…
In [B] Bowen defined the growth rate of an endomorphism of a finitely generated group and related it to the entropy of a map $f:M \mapsto M$ on a compact manifold. In this note we study the purely group theoretic aspects of the growth rate of an endomorphism of a finitely generated group. We show that it is finite and bounded by the maximum length of the image of a generator. An equivalent formulation is given that ties the growth rate of an endomorphism to an increasing chain of subgroups. We then consider the relationship between growth rate of an endomorphism on a whole group and the growth…
The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invariant. That involves proving that the rank of a wreath product of two symmetric groups is two and then use the fact that the endomorphism monoid of a partition is isomorphic to a wreath product of two full transformation semigroups. The calculation of the rank of these semigroups solves an open questio…
The aim of this note is to give a direct proof for the following result proved by Fountain and Lewin: {\em Let $\alg$ be an independence algebra of finite rank and let $a$ be a singular endomorphism of $\alg $. Then $a=e_1... e_n$ where $e^2_i=e_i$ and $rank(a)=rank(e_i)$.}
The new notion of adjoint algebraic entropy of endomorphisms of Abelian groups is introduced. Various examples and basic properties are provided. It is proved that the adjoint algebraic entropy of an endomorphism equals the algebraic entropy of the adjoint endomorphism of the Pontryagin dual. As applications, we compute the adjoint algebraic entropy of the shift endomorphisms of direct sums, and we prove an Addition Theorem for the adjoint algebraic entropy of bounded Abelian groups. A dichotomy is established, stating that the adjoint algebraic entropy of any endomorphism can take only values…
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.
For every finite-to-one map $λ:Γ\toΓ$ and for every abelian group $K$, the generalized shift $σ_λ$ of the direct sum $\bigoplus_ΓK$ is the endomorphism defined by $(x_i)_{i\inΓ}\mapsto(x_{λ(i)})_{i\inΓ}$. In this paper we analyze and compute the algebraic entropy of a generalized shift, which turns out to depend on the cardinality of $K$, but mainly on the function $λ$. We give many examples showing that the generalized shifts provide a very useful universal tool for producing counter-examples.
On the semigroup of monoid endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ with a two-element family $\mathscr{F}$ of inductive nonempty subsets of $ω$
Let $G$ be a Lie group over a totally disconnected local field and $α$ be an analytic endomorphism of $G$. The contraction group of $α$ ist the set of all $x\in G$ such that $α^n(x)\to e$ as $n\to\infty$. Call sequence $(x_{-n})_{n\geq 0}$ in $G$ an $α$-regressive trajectory for $x\in G$ if $α(x_{-n})=x_{-n+1}$ for all $n\geq 1$ and $x_0=x$. The anti-contraction group of $α$ is the set of all $x\in G$ admitting an $α$-regressive trajectory $(x_{-n})_{n\geq 0}$ such that $x_{-n}\to e$ as $n\to\infty$. The Levi subgroup is the set of all $x\in G$ whose $α$-orbit is relatively compact, and such t…
On the semigroup of endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$
We obtain conditions of uniform continuity for endomorphisms of free-abelian times free groups for the product metric defined by taking the prefix metric in each component and establish an equivalence between uniform continuity for this metric and the preservation of a coarse-median, which was recently introduced by Fioravanti. Considering the extension of an endomorphism to the completion we count the number of orbits for the action of the subgroup of fixed points (resp. periodic) points on the set of infinite fixed (resp. periodic) points. Finally, we study the dynamics of infinite points: f…
Centralisers of semi-simple elements are semidirect products
Université de Picardie-Jules Verne（亚眠皮卡第朱尔维尔纳大学）；LAMFA, CNRS UMR 7352（法国国家科学研究中心联合研究实验室 7352）；Université Paris Cité（巴黎西岱大学）；Sorbonne Université（索邦大学）；IMJ-PRG（数学与理论物理研究所）
In the quest in constructing conformal field theories (CFT) Jones has discovered a beautiful and deep connection between CFT, Richard Thompson's groups and knot theory. This led to a powerful functorial framework for constructing actions of particular groups arising from categories such as Thompson's groups and braid groups. In particular, given a group and two of its endomorphisms one can construct a semidirect product where the largest Thompson's group $V$ is acting. These semidirect products have remarkable diagrammatic descriptions which were previously used to provide new examples of grou…
We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $Γ$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $Γ$. Further structural information about End $R$ is established including that Aut $Γ$ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.
Let $G$ be a group and let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic zero. Denote $A=X(k)$ the set of rational points of $X$. We investigate invertible algebraic cellular automata $τ\colon A^G \to A^G$ whose local defining map is induced by some morphism of algebraic varieties $X^M \to X$ where $M$ is a finite memory. When $G$ is locally embeddable into finite groups (LEF), then we show that the inverses of reversible algebraic cellular automata are automatically algebraic cellular automata. Generalizations are also obtained for finite product Hopfian…
We give a characterization of those abelian groups which are direct sums of cyclic groups and the Jacobson radical of their endomorphism rings are closed. A complete characterization of $p$-groups $A$ for which $(EndA,\mathcal T_L)$ is locally compact, where $\mathcal T_L$ is the Liebert topology on $EndA$, is given. We prove that if $A$ is a countable elementary $p$-group then $EndA$ has a non-admissible ring topology. To every functorial topology on $A$ a right bounded ring topology on $EndA$ is attached. By using this topology we construct on $EndA$ a non-metrizable and non-admissibe ring t…
In this second article, we continue to study classes of groups constructed from a functorial method due to Vaughan Jones. A key observation of the author shows that these groups have remarkable diagrammatic properties that can be used to deduce their properties. Given any group and two of its endomorphisms, we construct a semidirect product. In our first article dedicated to this construction, we classify up to isomorphism all these semidirect products when one of the endomorphisms is trivial and described their automorphism group. In this article we focus on the case where both endomorphisms …
We prove a variety results on tensor product factorizations of finite dimensional Hopf algebras (more generally Hopf algebras satisfying chain conditions in suitable braided categories). The results are analogs of well-known results on direct product factorizations of finite groups (or groups with chain conditions) such as Fitting's Lemma and the uniqueness of the Krull-Remak-Schmidt factorization. We analyze the notion of normal (and conormal) Hopf algebra endomorphisms, and the structure of endomorphisms and automorphisms of tensor products. The results are then applied to compute the automo…
