A new proof of finiteness of maximal arithmetic reflection groups

TL;DR

This paper presents a novel proof demonstrating that there are finitely many maximal arithmetic reflection groups, avoiding traditional automorphic form tools by leveraging the arithmetic Margulis lemma.

Contribution

It introduces a new proof technique for finiteness that relies solely on the arithmetic Margulis lemma, bypassing trace formulas and automorphic form methods.

Findings

Proves finiteness of maximal arithmetic reflection groups

Develops a proof avoiding automorphic form tools

Utilizes the arithmetic Margulis lemma in a novel way

Abstract

We give a new proof of the finiteness of maximal arithmetic reflection groups. Our proof is novel in that it makes no use of trace formulas or other tools from the theory of automorphic forms and instead relies on the arithmetic Margulis lemma of Fraczyk, Hurtado and Raimbault.