The Lodha--Moore groups and their $n$-adic generalizations are not SCY

Abstract

A closed 4-manifold is symplectic Calabi--Yau (SCY) if its canonical class is trivial. Friedl and Vidussi proved that Thompson's group $F$ cannot be the fundamental group of any SCY manifold. In this paper, we show that its generalizations, called the Brown--Thompson group and the $n$-adic Lodha--Moore groups, cannot be also the fundamental group of any SCY manifold by using their method. From this proof, we also show that there exist non-trivial infinitely many examples which satisfy Geoghegan's conjecture.