Critical exponents of normal subgroups in higher rank

TL;DR

This paper investigates the relationship between the critical exponents of a discrete subgroup of a higher rank semi-simple Lie group and its Zariski dense normal subgroups, revealing inequalities and conditions for equality.

Contribution

It establishes that Zariski dense normal subgroups share the same limit cone as the original group and provides bounds on their critical exponents, with equality under amenability.

Findings

Limit cones of the subgroup and original group coincide.

Critical exponents of the subgroup are at least half those of the original group.

Equality of critical exponents occurs when the quotient is amenable.

Abstract

We study the critical exponents of discrete subgroups of a higher rank semi-simple real linear Lie group $G$. Let us fix a Cartan subspace $\mathfrak a\subset \mathfrak g$ of the Lie algebra of $G$. We show that if $\Gamma< G$ is a discrete group, and $\Gamma' \triangleleft \Gamma$ is a Zariski dense normal subgroup, then the limit cones of $\Gamma$ and $\Gamma'$ in $\mathfrak a$ coincide. Moreover, for all linear form $\phi : \mathfrak a\to \mathbb R$ positive on this limit cone, the critical exponents in the direction of $\phi$ satisfy $\displaystyle \delta_\phi(\Gamma') \geq \frac 1 2 \delta_\phi(\Gamma)$. Eventually, we show that if $\Gamma'\backslash \Gamma$ is amenable, these critical exponents coincide.