Lattices of Large Systole Containing a Fixed 3-Manifold Group

Paige Hillen (UCSB)

Abstract: One quantity which measures the geometry of a Riemannian manifold M is the systole: the minimal length of a non-contractible closed geodesic on M. On the other hand, a way to measure the topology of M is the systolic genus: the minimal genus such that a surface group of genus g is a subgroup of the fundamental group of M. For hyperbolic manifolds, the systolic genus is bounded from below in terms of the systole. However, this is no longer true for higher rank lattices. I will present recent work showing there exist infinitely many non-uniform lattices in SL(8,R) each with a sequence of commensurable lattices whose systole is going to infinity, yet they all contain the same 3-manifold group. As this 3-manifold group has surface subgroups, this condition is a stronger bound on the topological complexity, despite diverging geometric complexity. This is a notable characteristic of these lattices being higher rank.