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

TL;DR

This work extends Friedl–Vidussi’s obstruction against Thompson’s group $F$ being SCY to its generalizations, the Brown–Thompson groups $F(n)$ and the $n$-adic Lodha–Moore groups $G_0(n)$. By expressing these groups as strictly ascending HNN extensions and analyzing their cohomology $H^{i}(G;bZ[G])$, abelianizations, and $L^2$-Betti numbers, the authors show that for all $n\ge 2$, both $F(n)$ and $G_0(n)$ cannot be fundamental groups of SCY 4-manifolds (noting $b_1^{(2)}=0$ and $q(G)>0$). They further establish vanishing cohomology up to all degrees and apply a Hurewicz-at-infinity framework to prove trivial homotopy groups at infinity, yielding infinitely many examples that satisfy Geoghegan’s four conjectures. The results illuminate the scarcity of SCY 4-manifold groups and connect geometric-topological properties with ascending HNN-extensions in a broad class of Thompson-like groups.

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.

Figures (2)

• Figure 1: An example of a flat complex (before the one-point compactification).
• Figure 2: An example of $\mathcal{C}(\mathcal{A})$.

Theorems & Definitions (13)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof : Proof of Proposition \ref{['LM_nonFlike']}
• proof