Manifolds counting and class field towers

TL;DR

This paper disproves a conjecture about the growth rate of conjugacy classes of arithmetic lattices in certain Lie groups, showing it grows faster than previously conjectured, using class field towers with bounded root discriminant.

Contribution

It provides a counterexample to the conjectured growth rate and introduces the use of class field towers with bounded root discriminant in this context.

Findings

The growth rate of conjugacy classes is actually $x^{c\,\log x}$, not as previously conjectured.

Existence of towers of field extensions with bounded root discriminant is crucial for the proof.

Disproves the conjecture relating growth rate to the constant $\,\gamma(H)$.

Abstract

In [BGLM] and [GLNP] it was conjectured that if $H$ is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in $H$ of covolume at most $x$ is $x^{(\gamma(H)+o(1))\log x/\log\log x}$ where $\gamma(H)$ is an explicit constant computable from the (absolute) root system of $H$. In this paper we prove that this conjecture is false. In fact, we show that the growth is at rate $x^{c\log x}$. A crucial ingredient of the proof is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers.