Finiteness properties of some groups of piecewise projective homeomorphisms

Farley, Daniel S.

doi:10.2140/agt.2026.26.1395

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Finiteness properties of some groups of piecewise projective homeomorphisms

Daniel S. Farley

Algebraic & Geometric Topology 26 (2026) 1395–1449

DOI: 10.2140/agt.2026.26.1395

Abstract

The Lodha–Moore group $G$ is an $F_{\infty}$ counterexample to von Neumann’s conjecture. The group $G$ acts on the real line via piecewise projective homeomorphisms.

We will describe groups $F (S_{i})$ , $F (S_{i}^{'})$ , $T (S_{i})$ , $V (S_{i})$ , and $V (S_{i}^{'})$ for $i = 2$ and $3$ . All of these are groups of piecewise projective homeomorphisms that are modelled on Thompson’s groups $F$ , $T$ , and $V$ (respectively); each is “locally determined” by one of four inverse semigroups, which we denote by $S_{i}$ or $S_{i}^{'}$ ( $i = 2, 3$ ). Following a method developed by Hughes and the author, we will show that all ten groups have type $F_{\infty}$ .

The Lodha–Moore group $G$ is an ascending HNN extension of $F (S_{2}^{'})$ , and thus our results give a new proof that $G$ has type $F_{\infty}$ .

Keywords

generalized Thompson groups, Lodha–Moore groups, finiteness properties

Mathematical Subject Classification

Primary: 20F65, 20J05

Secondary: 20M18

References

Publication

Received: 5 March 2023

Revised: 14 June 2024

Accepted: 5 May 2025

Published: 25 April 2026

Authors

Daniel S. Farley
Department of Mathematics and Statistics Miami University Oxford, OH United States

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Volume 26, issue 4 (2026)