Simple algebraic groups with the same maximal tori, weakly commensurable Zariski-dense subgroups, and good reduction

TL;DR

This paper introduces a new criterion for good reduction of algebraic groups based on maximal tori, explores implications for finiteness conjectures and subgroup properties, and applies these ideas to specific group types and geometric contexts.

Contribution

It provides a novel characterization of good reduction using maximal tori, linking it to finiteness conjectures and subgroup rigidity phenomena in algebraic groups.

Findings

New criterion for good reduction based on maximal tori

Finiteness of the genus implied by the Finiteness Conjecture

Analysis of eigenvalue rigidity and applications to geometric structures

Abstract

We provide a new condition for an absolutely almost simple algebraic group to have good reduction with respect to a discrete valuation of the base field which is formulated in terms of the existence of maximal tori with special properties. This characterization, in particular, shows that the Finiteness Conjecture for forms of an absolutely almost simple algebraic group over a finitely generated field that have good reduction at a divisorial set of places of the field would imply the finiteness of the genus of the group at hand. It also leads to a new phenomenon that we refer to as "killing the genus by a purely transcendental extension." Yet another application deals with the investigation of "eigenvalue rigidity" of Zariski-dense subgroups, which in turn is related to the analysis of length-commensurable Riemann surfaces and general locally symmetric spaces. Finally, we analyze the Finiteness Conjecture and the genus problem for simple algebraic groups of type $\textsf{F}_4$.