Torsion homology growth and cycle complexity of arithmetic manifolds

Abstract

Let $M$ be an arithmetic hyperbolic $3$-manifold, such as a Bianchi manifold. We conjecture that there is a basis for the second homology of $M$, where each basis element is represented by a surface of “low” genus, and we give evidence for this. We explain the relationship between this conjecture and the study of torsion homology growth.