Homology growth in families of locally symmetric spaces

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Work of Serre, Ash, Calegari-Venkatesh, Bergeron-Venkatesh, and Scholze paints a very compelling picture of the significance of torsion in the homology of arithmetic groups. Systems of eigenvalues for Hecke operators acting on the homology of locally symmetric spaces associated with congruence arithmetic groups are now expected to be the universal source of number fields whose Galois groups are Lie groups of finite type. This expectation has been dubbed the "Langlands program over Z"; it includes Serre's conjecture on the modularity of odd, 2-dimensional mod p Galois representations as a special case. The Langlands Program over Z would be vacuous if not for quantitative (asymptotic) results proving that torsion in the homology of arithmetic groups is (at least sometimes) abundant. A convincing model predicting exactly which arithmetic groups should contain abundant torsion in their homology was laid out in work of Bergeron-Venkatesh; significant supporting evidence was proven ibid and in works of Marshall-Muller, Muller-Pfaff, and others. Current understanding of torsion growth, however, is very limited in "weights" for which both torsion and rational cohomology coexist. Furthermore, - If we expect torsion to be abundant in the homology of an explicit arithmetic group, then surely we should be able to "see it on a computer". - The landscape of reciprocity over Z is "wide open" and full of conjecture. The most germane conjectures therein are falsifiable and it seems very worthwhile to confirm them computationally, insofar as it is possible. State of the art approaches for computing the homology of arithmetic groups, however, are ad hoc and have limited scope due to algorithmic efficiency issues. Devising algorithms to efficiently compute the homology of arithmetic groups (and Hecke actions thereon) is therefore very worthwhile. My research program, over the next five years, will center around the above two themes. Namely, it will study: (A) growth of topological invariants in families of finite volume locally symmetric spaces, especially torsion in homology. (B) how to effectively and efficiently compute these invariants. My most significant progress on these problems: (A*) (from one year ago) Joint with Mark Stern, I show that tiny 1-form Laplacian eigenvalues on hyperbolic 3-manifolds, a known obstruction to growth of torsion in the first homology group, are related to the failure of short loops to be ``efficiently bounded." (B*) (ongoing) Joint with Aurel Page, we devise a general, efficient algorithm to computing the topology of congruence, arithmetic locally symmetric spaces and Hecke actions thereon. Torsion in the homology of arithmetic groups is a "hot topic"; the time is ripe for progress on problems (A) and (B) and such progress would have great utility. Meditation on my works with Stern and Page, alluded in (A*) and (B*), inspired many of specific problems I suggest in the present proposal.