BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Waist inequalities on groups and spaces - David Hume (University o
 f Birmingham)
DTSTART:20250717T104500Z
DTEND:20250717T114500Z
UID:TALK233581@talks.cam.ac.uk
DESCRIPTION:The waist inequality is a fundamental and deep result in Eucli
 dean geometry. It states that\, for any continuous map from the ball of ra
 dius $R$ in $\\mathbb{R}^n$ to $\\mathbb{R}^q$\, there is a point $z\\in\\
 mathbb{R}^q$ whose preimage is ``at least as big as'' the ball of radius $
 R$ in $\\mathbb{R}^{n-q}$. We may view it as a non-linear version of the r
 ank-nullity theorem.\nAnother way to view it\, and the one on which this t
 alk is based\, is as a measure of how "topologically expanding" balls in $
 \\mathbb{R}^n$ are. We use this perspective to define\, for each metric sp
 ace $X$\, a family of sublinear functions $\\mathrm{TO}^q_X:\\mathbb{N}\\t
 o\\mathbb{N}$. These functions satisfy a monotonicity property strikingly 
 similar to the growth function for finitely generated groups: for all (sui
 tably well-connected) metric spaces $X\,Y$\, whenever there is a coarse em
 bedding $X\\to Y$\, there is a constant $C$ such that\n\\[\n\\mathrm{TO}^q
 _X(r) \\leq C\\mathrm{TO}^q_Y(Cr)\n\\]\nContinuing the analogy with the gr
 owth function\, we identify two classes of function which seem to arise na
 turally (analogues of polynomial and exponential growth): $q$-thin\, $\\ma
 thrm{TO}^q_X(r)\\leq Cr^a$ for some $C\\geq 1$\, $a<1$\; and $q$-thick\, $
 \\mathrm{TO}^q_X(r)\\qeq cr/\\log(r)^a$ for some $c>0$\, $a>0$.\nTwo highl
 ights of the theory (as things stand) are: all finitely generated nilpoten
 t and hyperbolic groups (and all their fg subgroups) are $1$-thin (joint w
 ith Mackay-Tessera)\; and direct products of $n$ $3$-regular trees\, and r
 ank $n$ symmetric spaces of non-compact type are $n-1$-thick (using result
 s of Bensaid-Nguyen).&nbsp\;\n&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
END:VEVENT
END:VCALENDAR