HNN decompositions of the Lodha-Moore groups, and topological applications

TL;DR

This paper studies the Lodha-Moore groups, revealing their decompositions as ascending HNN extensions, and proves they have trivial homotopy groups at infinity, also computing key invariants related to their topological and algebraic properties.

Contribution

It provides new decompositions of Lodha-Moore groups and establishes their topological properties, including trivial homotopy groups at infinity and calculations of Bieri-Neumann-Strebel invariants.

Findings

Lodha-Moore groups have trivial homotopy groups at infinity.

Computed the Bieri-Neumann-Strebel invariant Sigma^1 and Sigma^2 for these groups.

Established decompositions into ascending HNN extensions, linking to Thompson's group F.

Abstract

The Lodha-Moore groups provide the first known examples of type F_\infty groups that are non-amenable and contain no non-abelian free subgroups. These groups are related to Thompson's group F in certain ways, for instance they contain it as a subgroup in a natural way. We exhibit decompositions of four Lodha-Moore groups, G, G_y, {_y}G and {_y}G_y, into ascending HNN extensions of isomorphic copies of each other, both in ways reminiscent to such decompositions for F and also in quite different ways. This allows us to prove two new topological results about the Lodha-Moore groups. First, we prove that they all have trivial homotopy groups at infinity; in particular they are the first examples of groups satisfying all four parts of Geoghegan's 1979 conjecture about F. Second, we compute the Bieri-Neumann-Strebel invariant Sigma^1 for the Lodha-Moore groups, and get some partial results for the Bieri-Neumann-Strebel-Renz invariants Sigma^m, including a full computation of Sigma^2.