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David Bruce Cohen

Publications and source records attributed to David Bruce Cohen.

10 recordsLinked to original sources
arXiv2018

Lamplighters admit weakly aperiodic SFTs

Let $A$ be a finite set and $G$ a group. A closed subset $X$ of $A^G$ is called a subshift if the action of $G$ on $A^G$ preserves $X$. If $K$ is a closed subset of $A^G$ such that membership in $K$ is determined by looking at a fixed finite set of coordinates, and $X$ is the intersection of all translates of $K$ under the action of $G$, then $X$ is called a subshift of finite type (SFT). If an SFT is nonempty and contains no finite $G$-orbits, it is said to be weakly aperiodic. A virtually cyclic group has no weakly aperiodic SFT, and Carroll and Penland have conjectured that a group with no weakly aperiodic SFT must be virtually cyclic. Answering a question of Jeandel, we show that lamplighters always admit weakly aperiodic SFTs.

math.GR↗
arXiv2017

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.

math.GR↗
arXiv2017

A counterexample to the easy direction of the geometric Gersten conjecture

For finitely generated groups H and G, equipped with word metrics, a translation-like action of H on G is a free action such that each element of H acts by a map which has finite distance from the identity map in the uniform metric. For example, if H is a subgroup of G, then right translation by elements of H yields a translation-like action of H on G. Whyte asked whether a group with no translation-like action by a Baumslag-Solitar group must be hyperbolic, where the free abelian group of rank 2 is understood to be a Baumslag-Solitar group. We show that the converse of this conjecture is false, and in particular the fundamental group of a closed hyperbolic 3-manifold admits a translation-like action by the free abelian group of rank 2.

math.GR↗
arXiv2017

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.

math.GR↗
arXiv2016

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.

math.GR↗
arXiv2015

Strongly aperiodic subshifts on surface groups

We give strongly aperiodic subshifts of finite type on every hyperbolic surface group; more generally, for each pair of expansive primitive symbolic substitution systems with incommensurate growth rates, we construct strongly aperiodic subshifts of finite type on their orbit graphs.

math.GR↗
arXiv2015

The large scale geometry of strongly aperiodic subshifts of finite type

A subshift on a group G is a closed, G-invariant subset of A^G, for some finite set A. It is said to be a subshift of finite type (SFT) if it is defined by a finite collection of 'forbidden patterns', to be strongly aperiodic if all point stabilizers are trivial, and weakly aperiodic if all point stabilizers are infinite index in G. We show that groups with at least 2 ends have a strongly aperiodic SFT, and that having such an SFT is a QI invariant for finitely presented torsion free groups. We show that a finitely presented torsion free group with no weakly aperiodic SFT must be QI-rigid. The domino problem on G asks whether the SFT specified by a given set of forbidden patterns is empty. We show that decidability of the domino problem is a QI invariant.

math.GR↗
arXiv2014

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.

math.AT↗
arXiv2014

The Dehn function of 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 is at least 5. By results of Broaddus, Farb, and Putman, this implies that the Torelli group in large genus is at most exponentially distorted.

math.GR↗