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

Part of: Low-dimensional topology Special aspects of infinite or finite groups

Published online by Cambridge University Press:  04 April 2022

YASH LODHA  and
MATTHEW C. B. ZAREMSKY
YASH LODHA
Affiliation:
Department of Mathematics, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria. e-mail: yashlodha763@gmail.com
MATTHEW C. B. ZAREMSKY
Affiliation:
Department of Mathematics and Statistics, University at Albany (SUNY), 1400 Washington Ave, Albany, NY 12222, U.S.A. e-mail: mzaremsky@albany.edu
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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 a question posed in Oberwolfach in 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.

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 57M07: Topological methods in group theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 1 , January 2023 , pp. 25 - 48
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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