Endomorphisms Of Groups Research Articles

Fixed points of endomorphisms of graph groups

Abstract.It is shown, for a given graph group

C*-Algebras Associated with Endomorphisms and Polymorphisms of Compact Abelian Groups

A surjective endomorphism or, more generally, a polymorphism in the sense of \cite{SV}, of a compact abelian group $H$ induces a transformation of $L^2(H)$. We study the C*-algebra generated by this operator together with the algebra of continuous functions $C(H)$ which acts as multiplication operators on $L^2(H)$. Under a natural condition on the endo- or polymorphism, this algebra is simple and can be described by generators and relations. In the case of an endomorphism it is always purely infinite, while for a polymorphism in the class we consider, it is either purely infinite or has a unique trace. We prove a formula allowing to determine the $K$-theory of these algebras and use it to compute the $K$-groups in a number of interesting examples.

Entropy of endomorphisms of Lie groups

We show, when $G$ is a nilpotent or reductive Lie group, that the entropy of any surjective endomorphism coincides with the entropy of its restriction to the toral component of the center of $G$. In particular, if $G$ is a semi-simple Lie group, the entropy of any surjective endomorphism always vanishes. Since every compact group is reductive, the formula for the entropy of a endomorphism of a compact group reduces to the formula for the entropy of an endomorphism of a torus. We also characterize the recurrent set of conjugations of linear semi-simple Lie groups.

STRING NUMBERS OF ABELIAN GROUPS

The string number of self-maps arose in the context of algebraic entropy and it can be viewed as a kind of combinatorial entropy function. Later on, its values for endomorphisms of abelian groups were calculated in full generality. We study its global version for abelian groups, providing several examples involving also Hopfian abelian groups. Moreover, we characterize the class of all abelian groups with string number zero in many cases and discuss its stability properties.

Entropy for endomorphisms of LCA groups

We introduce a modified version of the entropy defined for locally compact Abelian groups by Peters. This approach allows us to work with endomorphisms instead of working with automorphisms. We study some of the basic properties of this new entropy and we give direct proofs of the formulae that allow one to compute the entropy of endomorphisms of ZN, RN and CN, for every positive integer N.

Adjoint entropy vs topological entropy

Recently the adjoint algebraic entropy of endomorphisms of abelian groups was introduced and studied in Dikranjan (2010) [6]. We generalize the notion of adjoint entropy to continuous endomorphisms of topological abelian groups. Indeed, the adjoint algebraic entropy is defined using the family of all finite-index subgroups, while we take only the subfamily of all open finite-index subgroups to define the topological adjoint entropy. This allows us to compare the topological adjoint entropy with the known topological entropy of continuous endomorphisms of compact abelian groups. In particular, the topological adjoint entropy and the topological entropy coincide on continuous endomorphisms of totally disconnected compact abelian groups. Moreover, we prove two so-called Bridge Theorems between the topological adjoint entropy and the algebraic entropy using respectively the Pontryagin duality and the precompact duality.

Bounds for fixed points and fixed subgroups on surfaces and graphs

We consider selfmaps of hyperbolic surfaces and graphs, and give some bounds involving the rank and the index of fixed point classes. One consequence is a rank bound for fixed subgroups of surface group endomorphisms, similar to the Bestvina‐Handel bound (originally known as the Scott conjecture) for free group automorphisms. When the selfmap is homotopic to a homeomorphism, we rely on Thurston’s classification of surface automorphisms. When the surface has boundary, we work with its spine, and Bestvina‐Handel’s theory of train track maps on graphs plays an essential role. It turns out that the control of empty fixed point classes (for surface automorphisms) presents a special challenge. For this purpose, an alternative definition of fixed point class is introduced, which avoids covering spaces hence is more convenient for geometric discussions. 55M20, 57M07; 20F34, 57M15, 57N05

𝔬K0-quasi-abelian varieties with complex multiplication

We are interested in finding the relation between rings of endomorphisms of toroidal groups and algebraic number fields which are not totally real. For algebraic number fields K , we define real maximal algebraic number fields K 0 and 𝔬 K 0 -quasi-abelian varieties. We show that any simple 𝔬 K 0 -quasi-abelian variety is constructed by embeddings of a non-totally real algebraic number field K and it has finite dimensional cohomology groups.

STRINGS OF GROUP ENDOMORPHISMS

Recently the strings and the string number of self-maps were used in the computation of the algebraic entropy of specific abelian group endomorphisms. We introduce two special kinds of strings, and their relative string numbers. We show that a dichotomy holds for all these three string numbers; in fact, they admit only zero and infinity as values on abelian group endomorphisms.

Adjoint algebraic entropy

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 zero or infinity. As a consequence, we obtain the following surprising discontinuity criterion for endomorphisms: every endomorphism of a compact abelian group, having finite positive algebraic entropy, is discontinuous.

The twisted conjugacy problem for endomorphisms of polycyclic groups

An algorithm is constructed that, when given an explicit presentation of a polycyclic group G, decides for any endomorphism ψ ∈ End(G) and any pair of elements u, v ∈ G, whether or not the equation (xψ)u = vx has a solution x ∈ G. Thus it is shown that the problem of the title is decidable. Also we present an algorithm that produces a finite set of generators of the subgroup Fixψ(G) ⩽ G of all ψ-invariant elements of G.

A generalization of a theorem by Bigard–Keimel and Conrad–Diem

Let G be an archimedean l-group and \({\mathfrak{P}(G)}\) denote the set of all polar preserving bounded group endomorphisms of G. Bigard and Keimel in [Bull. Soc. Math. France 97 (1969), 381–398] and, independently, Conrad and Diem in [Illinois J. Math. 15 (1971), 222–240] proved that \({\mathfrak{P}(G)}\) is an archimedean l-group with respect to the pointwise addition and ordering. This classical result is extended in this paper to certain sets of disjointness preserving bounded homomorphisms on archimedean l-groups.

Dirichlet series for finite combinatorial rank dynamics

We introduce a class of group endomorphisms -- those of finite combinatorial rank -- exhibiting slow orbit growth. An associated Dirichlet series is used to obtain an exact orbit counting formula, and in the connected case this series is shown to have a closed rational form. Analytic properties of the Dirichlet series are related to orbit-growth asymptotics: depending on the location of the abscissa of convergence and the degree of the pole there, various orbit-growth asymptotics are found, all of which are polynomially bounded.

Algebraic entropy for Abelian groups

The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. Here we study the algebraic entropy of the endomorphisms of Abelian groups, introduced in 1965 by Adler, Konheim and McAndrew. The so-called Addition Theorem is proved; this expresses the algebraic entropy of an endomorphism φ of a torsion group as the sum of the algebraic entropies of the restriction to a φ-invariant subgroup and of the endomorphism induced on the quotient group. Particular attention is paid to endomorphisms with zero algebraic entropy as well as to groups all of whose endomorphisms have zero algebraic entropy. The significance of this class arises from the fact that any group not in this class can be shown to have endomorphisms of infinite algebraic entropy, and we also investigate such groups. A uniqueness theorem for the algebraic entropy of endomorphisms of torsion Abelian groups is proved.

The twisted conjugacy problem for endomorphisms of metabelian groups

Let M be a finitely generated metabelian group explicitly presented in a variety $ {\mathcal{A}}^2 $ of all metabelian groups. An algorithm is constructed which, for every endomorphism φ ∈ End(M) identical modulo an Abelian normal subgroup N containing the derived subgroup M′ and for any pair of elements u, v ∈ M, decides if an equation of the form (xφ)u = vx has a solution in M. Thus, it is shown that the title problem under the assumptions made is algorithmically decidable. Moreover, the twisted conjugacy problem in any polycyclic metabelian group M is decidable for an arbitrary endomorphism φ ∈ End(M).

Polynomial Maps over p -Adics and Residual Properties of Mapping Tori of Group Endomorphisms

We continue our study of residual properties of mapping tori of free group endomorphisms. In this paper, we prove that each of these groups are virtually residually (finite p)-groups for all but finitely many primes p. The method involves further studies of polynomial maps over finite fields and p-adic completions of number fields.

ENDOMORPHISMS OF RELATIVELY HYPERBOLIC GROUPS

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a nonelementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out (G) is infinite, then G splits over a slender group. • If H is a nonparabolic subgroup of a relatively hyperbolic group, and if any isometric H-action on an ℝ-tree is trivial, then H is Hopfian. • If G is a nonelementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a nonelementary relatively hyperbolic quotient that is Hopfian. • Any finitely presented group is isomorphic to a finite index subgroup of Out (H) for some group H with Kazhdan property (T). (This sharpens a result of Ollivier–Wise).

Lehmer's Question, Knots and Surface Dynamics

AbstractLehmer's question is equivalent to one about generalized growth rates of Lefschetz numbers of iterated pseudo-Anosov surface homeomorphisms. One need consider only homeomorphisms that arise as monodromies of fibered knots in lens spacesL(n, 1),n> 0. Lehmer's question for Perron polynomials is equivalent to one about generalized growth rates of words under free group endomorphisms.

Virtual endomorphisms of nilpotent groups

A virtual endomorphism of a group G is a homomorphism f : H→ G where H is a subgroup of G of finite index m . The triple (G,H,f) produces a state-closed (or, self-similar) representation φ of G on the 1 -rooted m -ary tree. This paper is a study of properties of the image G^φ when G is nilpotent. In particular, it is shown that if G is finitely generated, torsion-free and nilpotent then G^φ has solvability degree bounded above by the number of prime divisors of m .

Schur multipliers and power endomorphisms of groups

We obtain new bounds for the exponent of the Schur multiplier of a given p-group. We prove that the exponent of the Schur multiplier can be bounded by a function depending only on the exponent of a given group. As a consequence we show that the exponent of the Schur multiplier of any group of exponent four divides eight, and that this bound is best possible. The notion of the exponential rank of a p-group is introduced. We show that powerful p-groups have exponential rank either zero or one.