Growth of cohomology of arithmetic groups and the stable trace formula: the case of $U(2,1)$

TL;DR

This paper develops a method using the stable trace formula to analyze the growth of cohomology in arithmetic groups and applies it specifically to the case of $U(2,1)$, confirming previous results.

Contribution

It introduces a strategy leveraging the stable trace formula for cohomology growth and demonstrates its application to $U(2,1)$, recovering known results.

Findings

Confirmed the growth behavior of cohomology for $U(2,1)$

Validated the use of the stable trace formula in this context

Reproduced Marshall's results using new approach

Abstract

We present a strategy to use the stable trace formula to compute growth in the cohomology of cocompact arithmetic lattices in a reductive group $G$. We then implement it in the case where $G = U(2,1)$, using results of Rogwaski. This recovers a result of Marshall.