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

Abstract A closed symplectic 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 4-manifold. In this paper, we show that its generalizations, called the Brown–Thompson groups, and the 𝑛-adic Lodha–Moore groups, also cannot be the fundamental group of any SCY 4-manifold by using their method. From this proof, we also show that there exist infinitely many non-trivial examples which satisfy Geoghegan’s conjecture.