On approximation properties of semidirect products of groups

Let R be a class of groups closed under taking semidirect products with finite kernel and fully residually R-groups. We prove that R contains all R-by-{finitely generated residually finite} groups. It follows that a semidirect product of a finitely generated residually finite group with a surjunctive group is surjunctive. This remained unknown even for direct products of a surjunctive group with the integers Z.Comment: 6 pages, no figure

Uniform symplicity of groups with proximal action

We prove that groups acting boundedly and order-primitively on linear orders or acting extremly proximality on a Cantor set (the class including various Higman-Thomson groups and Neretin groups of almost automorphisms of regular trees, also called groups of spheromorphisms) are uniformly simple. Explicit bounds are provided.Comment: 23 pages, appendix by Nir Lazarovich, corrected versio

Cancelation norm and the geometry of biinvariant word metrics

We study biinvariant word metrics on groups. We provide an efficient algorithm for computing the biinvariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the biinvariant Cayley graph of a nonabelian free group. We investigate the geometry of cyclic subgroups. We observe that in many classes of groups cyclic subgroups are either bounded or detected by homogeneous quasimorphisms. We call this property the bq-dichotomy and we prove it for many classes of groups of geometric origin.Comment: 32 pages, to appear in Glasgow Journal of Mathematic

Odd-dimensional Charney-Davis conjecture

More than once we have heard that the Charney-Davis Conjecture makes sense only for odd-dimensional spheres. This is to point out that in fact it is also a statement about even-dimensional spheres.Comment: 3 pages, no figure

On Normal Subgroups of Coxeter Groups Generated by Standard Parabolic Subgroups

We discuss one construction of nonstandard subgroups in the category of Coxeter groups. Two formulae for the growth series of such a subgroups are given. As an application we construct a flag simple convex polytope, whose f-polynomial has non-real roots.Comment: 12 pages, figure

Finite index subgroups in Chevalley groups are bounded : an addendum to "On bi-invariant word metrics"

Acknowledgement. The work was partially funded by a Leverhulme Trust Research Project Grant RPG-2017-159 and the Polish National Science Centre grant 2017/27/B/ST1/01467.Peer reviewedPostprin