The waist inequality is a fundamental and deep result in Euclidean geometry. It states that, for any continuous map from the ball of radius $R$ 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 rank-nullity theorem.
Another way to view it, and the one on which this talk 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 space $X$, a family of sublinear functions $\mathrm{TO}^q_X:\mathbb{N}\to\mathbb{N}$. These functions satisfy a monotonicity property strikingly similar to the growth function for finitely generated groups: for all (suitably well-connected) metric spaces $X,Y$, whenever there is a coarse embedding $X\to Y$, there is a constant $C$ such that
\[
\mathrm{TO}^q_X(r) \leq C\mathrm{TO}^q_Y(Cr)
\]
Continuing the analogy with the growth function, we identify two classes of function which seem to arise naturally (analogues of polynomial and exponential growth): $q$-thin, $\mathrm{TO}^q_X(r)\leq Cr^a$ for some $C\geq 1$, $a<1$; and $q$-thick, $\mathrm{TO}^q_X(r)\qeq cr/\log(r)^a$ for some $c>0$, $a>0$.
Two highlights of the theory (as things stand) are: all finitely generated nilpotent and hyperbolic groups (and all their fg subgroups) are $1$-thin (joint with Mackay-Tessera); and direct products of $n$ $3$-regular trees, and rank $n$ symmetric spaces of non-compact type are $n-1$-thick (using results of Bensaid-Nguyen).
