The BNSR-invariants of the Lodha-Moore groups, and an exotic simple group of type $\textrm{F}_\infty$

Abstract

In this paper we give a complete description of the Bieri-Neumann-Strebel-Renz invariants of the Lodha-Moore groups. The second author previously computed the first two invariants, and here we show that all the higher invariants coincide with the second one, which finishes the complete computation. As a consequence, we present a complete picture of the finiteness properties of normal subgroups of the first Lodha-Moore group. In particular, we show that every finitely presented normal subgroup of the group is of type $\textrm{F}_\infty$, answering question 112 from Oberwolfach Rep., 15(2):1579-1633, 2018. The proof involves applying a variation of Bestvina-Brady discrete Morse theory to the so called cluster complex $X$ introduced by the first author. As an application, we also demonstrate that a certain simple group $S$ previously constructed by the first author is of type $\textrm{F}_\infty$. This provides the first example of a type $\textrm{F}_\infty$ simple group that acts faithfully on the circle by homeomorphisms, but does not admit any nontrivial action by $C^1$-diffeomorphisms, nor by piecewise linear homeomorphisms, on any $1$-manifold.