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.