Arbitrarily large families of spaces of the same volume

TL;DR

This paper demonstrates the existence of arbitrarily large families of non-isomorphic arithmetic lattices sharing the same covolume within certain semi-simple Lie groups, expanding understanding of lattice diversity.

Contribution

It introduces a method to construct large families of non-isomorphic lattices with identical covolume using Bruhat-Tits theory and Prasad's volume formula.

Findings

Existence of arbitrarily large families of such lattices.

Construction method based on Bruhat-Tits theory.

Control of covolumes via Prasad's volume formula.

Abstract

In any connected non-compact semi-simple Lie group without factors locally isomorphic to SL_2(R), there can be only finitely many lattices (up to isomorphism) of a given covolume. We show that there exist arbitrarily large families of pairwise non-isomorphic arithmetic lattices of the same covolume. We construct these lattices with the help of Bruhat-Tits theory, using Prasad's volume formula to control their covolumes.