David Bruce Cohen

A Dehn function for Sp(2n, ℤ)

Gromov conjectured that any irreducible lattice in a symmetric space of rank at least 3 should have at most polynomial Dehn function. We prove that the lattice Sp(2p; ℤ) has quadratic Dehn function when p ≥ 5. By results of Broaddus, Farb, and Putman, this implies that the Torelli group in large genus is at most exponentially distorted.

Lipschitz 1-connectedness for some solvable Lie groups

A space X is said to be Lipschitz 1-connected if every L-Lipschitz loop in X bounds a O(L)-Lipschitz disk. A Lipschitz 1-connected space admits a quadratic isoperimetric inequality, but it is unknown whether the converse is true. Cornulier and Tessera showed that certain solvable Lie groups have quadratic isoperimetric inequalities, and we extend their result to show that these groups are Lipschitz 1-connected.

Continuous Cocycle Superrigidity for the Full Shift Over a Finitely Generated Torsion Group

Chung and Jiang showed that, if a one ended group contains an infinite order element, then every continuous cocycle over the full shift on that group, taking values in a discrete group, must be cohomologous to a homomorphism. We show that their conclusion holds for all one ended groups, so that the hypothesis of admitting an infinite order element may be omitted.

Translation-like actions of nilpotent groups

We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other.

Borel stability for congruence subgroups

We prove a homological stability theorem for congruence subgroups of symplectic groups. From this theorem, we deduce a generalization of a theorem of Borel showing that certain homology groups of a congruence subgroup do not depend on the level of the congruence subgroup.