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In this paper we give a complete description of theBieri-Neumann-Strebel-Renz invariants of the Lodha-Moore groups. The secondauthor previously computed the first two invariants, and here we show that allthe higher invariants coincide with the second one, which finishes the completecomputation. As a consequence, we present a complete picture of the finitenessproperties 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 ofBestvina-Brady discrete Morse theory to the so called cluster complex $X$introduced by the first author. As an application, we also demonstrate that acertain 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 byhomeomorphisms, but does not admit any nontrivial action by$C^1$-diffeomorphisms, nor by piecewise linear homeomorphisms, on any$1$-manifold.

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