Upper bounds for the entropy in the cusp for one-parameter diagonal flows on $SL_{d}(\mathbb{R})/SL_{d}(\mathbb{Z})$

TL;DR

This paper establishes explicit upper bounds for the entropy in the cusp for one-parameter diagonal flows on the space of lattices, with bounds tailored to different parabolic subgroups, and plans to prove their tightness in future work.

Contribution

It provides the first explicit upper bounds for cusp entropy in this setting, using a novel method involving auxiliary linear functionals on the Lie algebra.

Findings

Explicit upper bounds for cusp entropy are derived.

Bounds are provided for the entire cusp and specific parabolic regions.

Future work will prove these bounds are tight.

Abstract

We give explicit upper bounds for the entropy in the cusp for one-parameter diagonal flows on $SL_d(\mathbb{R})/SL_{d}(\mathbb{Z})$. These results include bounds for the entropy of the cusp as a whole, as well as for the cusp regions corresponding to either the maximal parabolic subgroups of $SL_{d}$, or the minimal (Borel) parabolic subgroup. To do so, we use a method which involves choosing an auxiliary linear functional on the Lie algebra of the Cartan group, specifically tailored to each of the bounds we are interested in. In a follow-up paper we prove that the upper bounds we obtain in this work are tight.