Minimal entropy for uniform lattices in PSL$_2(\mathbb{R})^n$

TL;DR

This paper proves that for compact quotients of the product of hyperbolic planes with fixed volume, the hyperbolic metric product minimizes volume entropy, highlighting a minimal entropy property in geometric structures.

Contribution

It establishes the minimal entropy property of hyperbolic metrics among all metrics with the same volume on compact quotients of hyperbolic plane products.

Findings

Hyperbolic metric product minimizes volume entropy

Minimal entropy property holds for fixed volume quotients

Results extend understanding of entropy in hyperbolic geometry

Abstract

We prove that, among metrics on a compact quotient of $\left(\mathbb{H}^2\right)^n$ (product of hyperbolic planes) of prescribed total volume, the product of hyperbolic metrics has minimal volume entropy.