Snowflakes, cones, and shortcuts

Christopher Cashen (Vienna)

22-Nov-2022, 14:00-16:00 (16 months ago)

Abstract: A graph is strongly shortcut if there exists \(K>1\) and a bound on the length of \(K\)-biLipschitz embedded cycles. A group is strongly shortcut if it acts geometrically on a strongly shortcut graph. This is a kind of non-positive curvature condition enjoyed by hyperbolic and CAT(0) groups, for example. Strongly shortcut groups are finitely presented and have all of their asymptotic cones simply connected (so have polynomial Dehn function).

We look at an infinite family of snowflake groups, which are known to have polynomial Dehn function, and show that all of their asymptotic cones are simply connected. The usual ways to show that a group has all asymptotic cones simply connected are to show that it is either of polynomial growth or has quadratic Dehn function, but our groups have neither of these properties. We also show that the 'obvious' Cayley graph is not strongly shortcut. This implies that some of its asymptotic cones contain isometrically embedded circles, so they have metrically nontrivial loops even though there are no topologically nontrivial loops. Here are two questions:

1. If a group has all of its asymptotic cones simply connected, does that imply that it is strongly shortcut?

2. Is it true that one Cayley graph of a group is strongly shortcut if and only if every Cayley graph of that group is strongly shortcut?

Our snowflake examples show that the answer to one of these questions is 'no'.

This is joint work with Nima Hoda and Daniel Woohouse.

algebraic topologyfunctional analysisgroup theorygeometric topologyoperator algebras

Audience: researchers in the topic


Vienna Geometry and Analysis on Groups Seminar

Organizer: Christopher Cashen*
*contact for this listing

Export talk to