Let be an arithmetic hyperbolic -manifold, such as a Bianchi manifold. We conjecture that there is a basis for the second homology of , 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.
"Torsion homology growth and cycle complexity of arithmetic manifolds." Duke Math. J. 165 (9) 1629 - 1693, 15 June 2016. https://doi.org/10.1215/00127094-3450429