Finiteness properties of some groups of piecewise projective homeomorphisms
Daniel Scott Farley · Algebraic & Geometric Topology · 2026
The Lodha-Moore group G is an F 1 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 0 i /, T .S i /, V .S i /, and V .S 0 i / for i D 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 0 i (i D 2; 3).Following a method developed by Hughes and the author, we will show that all ten groups have type F 1 .The Lodha-Moore group G is an ascending HNN extension of F.S 0 2 /, and thus our results give a new proof that G has type F 1 .1412 7. Finite complete presentations of semigroups 1421 8.An intermediate value theorem for the expansion scheme E i 1433 9.The proof of the F 1 property