On the $\Sigma$-invariants of wreath products

TL;DR

This paper provides a comprehensive description of the Bieri-Neumann-Strebel invariant for restricted permutational wreath products and explores related homotopical invariants, with applications to automorphism Reidemeister numbers.

Contribution

It offers the first complete characterization of these invariants for wreath products, extending understanding of their algebraic and topological properties.

Findings

Full description of the Bieri-Neumann-Strebel invariant for wreath products.

Partial results on the 2-dimensional homotopical invariant.

Applications to Reidemeister number of automorphisms.

Abstract

We present a full description of the Bieri-Neumann-Strebel invariant of restricted permutational wreath products of groups. We also give partial results about the 2-dimensional homotopical invariant of such groups. These results may be turned into a full picture of these invariants when the abelianization of the basis group is infinite. We apply these descriptions to the study of the Reidemeister number of automorphisms of wreath products in some specific cases.