On the growth of the first Betti number of arithmetic hyperbolic 3-manifolds

TL;DR

This paper establishes a lower bound on the growth of the first Betti number in certain arithmetic hyperbolic 3-manifolds, showing it increases at least as fast as the square root of the subgroup index.

Contribution

It provides a novel lower bound for the first Betti number growth in arithmetic hyperbolic 3-manifolds using Lefschetz number calculations and Galois automorphisms.

Findings

First Betti number grows at least as fast as the square root of the subgroup index.

Lower bounds are derived for Betti numbers in specific arithmetic hyperbolic 3-manifolds.

The approach involves Lefschetz number calculations and Galois automorphisms.

Abstract

We calculate the Lefschetz number of a Galois automorphism in the cohomology of certain arithmetic congruence groups arising from orders in quaternion algebras over number fields. As an application we give a lower bound for the first Betti number of a class of arithmetically defined hyperbolic 3-manifolds and we deduce the following theorem: Given an arithmetically defined cocompact subgroup of SL(2,C), provided the underlying quaternion algebra meets some conditions, there is a decreasing sequence of finite index subgroups of such that the first Betti number grows at least as fast as the square root of the index.