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

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.