Algebraic entropy on topologically quasihamiltonian groups

Abstract

We study the algebraic entropy of continuous endomorphisms of compactly covered, locally compact, topologically quasihamiltonian groups. We provide a Limit-free formula which helps us to simplify the computations of this entropy. Moreover, several Addition Theorems are given. In particular, we prove that the Addition Theorem holds for endomorphisms of quasihamiltonian torsion FC-groups (e.g., Hamiltonian groups).2010 Mathematics Subject Classification. 37A35, 22D40, 28D20, 20K35.

Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups

“…Clearly, this result extends a consequence of [14, Corollary 7.2] stating that AT .G; ; H / holds for every torsion F C -group G, every 2 End.G/ and every -invariant normal subgroup H of G with H surjective and N G=H injective. Moreover, it extends one of the main results from [39], namely, that AT .G/ holds for every locally finite group G which is a quasihamiltonian F C -group. Indeed, Theorem 1.4 covers the family (2) in the above diagram, and Corollary 1.5 the union of (3) and (4), while the result from [14] concerns the groups in (4), and the result from [39] the groups in the intersection of (3) and (4).…”

“…(c) In [39], the following example was given of a quasihamiltonian locally finite group H that is not an F C -group; this means that H is in (3), but it is not in (4). Let H D Z N 3 2 Ì˛Z 3 , where˛is the action of Z 3 on Z N 9 defined, for every x 2 Z 3 and every a 2 Z N 9 , by˛.x/.a/ D 4 x a.…”

“…Let H D Z N 3 2 Ì˛Z 3 , where˛is the action of Z 3 on Z N 9 defined, for every x 2 Z 3 and every a 2 Z N 9 , by˛.x/.a/ D 4 x a. Since H is a non-abelian 3-group that satisfies [20,Theorem 3], so H is quasihamiltonian; but H 0 is infinite, and so H cannot be an F C -group by [39,Proposition 2.8].…”

“…In another direction the algebraic entropy was recently extended to left actions of cancellative right amenable semigroups on abelian groups in [4] (see also the work of Virili [38] with applications to Kaplansky's Stable Finiteness Conjecture and Zero-Divisors Conjecture). Moreover, we refer the interested reader to [15,31,37,41] for results on the algebraic entropy for continuous endomorphisms of locally compact groups, and to [7,10,14,39,40] for the connection of the algebraic entropy with the topological entropy via Pontryagin duality. Now we give the definition of algebraic entropy in the setting we are interested in, that is, for group endomorphisms.…”

“…(d) We give reference to [14,29,35,39] for results on algebraic entropy for continuous endomorphisms of locally compact groups, and to [6,10,13,37,38] for the connection of algebraic entropy with topological entropy via Pontryagin duality.…”