Cocompact lattices of minimal covolume in rank 2 Kac-Moody groups, Part II

TL;DR

This paper establishes the minimal covolume for cocompact lattices in rank 2 Kac-Moody groups over finite fields and constructs an explicit lattice achieving this bound, completing previous work on edge-transitive cases.

Contribution

It provides the first explicit lower bound and construction of minimal covolume cocompact lattices in rank 2 Kac-Moody groups over finite fields.

Findings

Determined a positive lower bound on covolumes of cocompact lattices.

Constructed a cocompact lattice achieving the minimal covolume.

Completed the classification for the minimal covolume lattices in the specified groups.

Abstract

Let G be a topological Kac-Moody group of rank 2 with symmetric Cartan matrix, defined over a finite field F_q. An example is G = SL(2,F_q((t^{-1}))). We determine a positive lower bound on the covolumes of cocompact lattices in G, and construct a cocompact lattice \Gamma_0 < G which realises this minimum. This completes the work begun in Part I, which considered the cases when G admits an edge-transitive lattice.