On finite extensions of lamplighter groups

TL;DR

This paper investigates extensions of lamplighter groups, revealing complex properties such as decidability issues, rational growth series, and language classifications, thereby addressing key open questions in combinatorial group theory.

Contribution

It introduces new groups extending lamplighter groups that demonstrate diverse combinations of decidability and complexity properties, advancing understanding in the field.

Findings

Decidable subgroup membership with undecidable uniform subgroup membership problem

Rational volume growth series with undecidable word problem

Recursive language of conjugacy geodesics with decidable word problem and undecidable conjugacy problem

Abstract

We study a family of groups consisting of the simplest extensions of lamplighter groups. We use these groups to answer multiple open questions in combinatorial group theory, providing groups that exhibit various combinations of properties: 1) Decidable Subgroup Membership and undecidable Uniform Subgroup Membership Problem, 2) Rational volume growth series and undecidable Word Problem and 3) Recursive (even context-free) language of conjugacy geodesics, decidable Word Problem, and undecidable Conjugacy Problem. We also consider the co-Word Problem, residual finiteness and the Isomorphism Problem within this class.