BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Christopher Cashen (Vienna) DTSTART:20221122T140000Z DTEND:20221122T160000Z DTSTAMP:20260423T041008Z UID:WienGAGT/31 DESCRIPTION:Title: Snowflakes\, cones\, and shortcuts\nby Christopher Cashen (Vienna) a s part of Vienna Geometry and Analysis on Groups Seminar\n\n\nAbstract\nA graph is strongly shortcut if there exists \\(K>1\\) and a bound on the le ngth of \\(K\\)-biLipschitz embedded cycles. A group is strongly shortcut if it acts geometrically on a strongly shortcut graph. This is a kind of n on-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 functi on).\n\n We look at an infinite family of snowflake groups\, which are kno wn to have polynomial Dehn function\, and show that all of their asymptoti c 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 polynom ial growth or has quadratic Dehn function\, but our groups have neither of these properties. We also show that the 'obvious' Cayley graph is not str ongly shortcut. This implies that some of its asymptotic cones contain iso metrically embedded circles\, so they have metrically nontrivial loops eve n though there are no topologically nontrivial loops. Here are two questio ns:\n\n 1. If a group has all of its asymptotic cones simply connected\, d oes that imply that it is \nstrongly shortcut? \n\n2. Is it true that one Cayley graph of a group is strongly shortcut if and only if every Cayley g raph of that group is strongly shortcut? \n\nOur snowflake examples show t hat the answer to one of these questions is 'no'. \n\nThis is joint work w ith Nima Hoda and Daniel Woohouse.\n LOCATION:https://researchseminars.org/talk/WienGAGT/31/ END:VEVENT END:VCALENDAR