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

TL;DR

This paper proves that certain complex groups, including the Brown--Thompson and $n$-adic Lodha--Moore groups, cannot be fundamental groups of symplectic Calabi--Yau 4-manifolds, extending previous results about Thompson's group F.

Contribution

It extends the non-existence results for fundamental groups of SCY manifolds to a broader class of groups, using methods from prior work on Thompson's group.

Findings

Brown--Thompson groups are not fundamental groups of SCY manifolds.

$n$-adic Lodha--Moore groups are not fundamental groups of SCY manifolds.

Existence of infinitely many groups satisfying Geoghegan's conjecture.

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.