This paper is concerned with the entropy of an action of a countable torsion-free abelian group G by continuous automorphisms of a compact abelian group X. A formula is obtained that expresses the entropy in terms of the Mahler measure of a greatest common divisor, complementing earlier work by Einsiedler, Lind, Schmidt and Ward. This leads to a uniform method for calculating entropy whenever G...

We prove new separability results about free groups. Namely, if H1,…,Hk are infinite index, finitely generated subgroups of a non-abelian group F, then there exists homomorphism onto some alternating f:F↠Am such that whenever Hi is not conjugate into Hj, f(Hi) f(Hj). The proof probabilistic. count the expected number fixed points f(Hi)'s and their under carefully constructed measure.

this work is a continuation of [a‎. ‎o‎. ‎asar‎, ‎‎‎on infinitely generated groups whose proper subgroups are solvable‎, ‎{em j‎. ‎algebra}‎, ‎{bf 399} (2014) 870-886.]‎, ‎where it was shown‎ ‎that a perfect infinitely generated group whose proper subgroups‎ are solvable and in whose homomorphic images normal closures of ‎finitely generated subgroups are residually nilpotent is a fitting‎‎$p$-g...

We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products and abelian groups: after introducing enriched automata (i.e., with extra labels), we obtain an explicit bijection between a certain type such automata, which - it happens in group is computable finitely generated case. This approach provides neat geometric description (even non gener...

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorp...

Let k[G] be the group algebra, where G is a finite abelian p-group and k is a field of characteristic p. A complete classification of finitely generated k[G]-modules is available only when G is cyclic, Cpn , or C2 × C2. Tackling the first interesting case, namely modules over k[C2 × C4], some structure theorems revealing the differences between elementary and non-elementary abelian group cases ...

Abstract We show that the commensurator of any finitely generated abelian subgroup H in a biautomatic group centralises finite-index . deduce CAT(0) groups introduced by Leary and Minasyan (Commensurating HNN-extensions: non-positive curvature biautomaticity, Geom Topol 25 :1819–1860, 2021) are either or cannot arise as subgroups groups, answering question posed generalising an analogous result...

Abstract We study the quasi-isometric rigidity of a large family finitely generated groups that split as graphs with virtually free vertex and two-ended edge groups. Let G be group is one-ended, hyperbolic relative to abelian subgroups, has JSJ decomposition over subgroups containing only are not quadratically hanging. Our main result any abstractly commensurable . In particular, our applies ce...