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In the last set of notes, we obtained the following structural theorem concerning approximate groups:

Theorem 1 Let ${A}$ be a finite ${K}$-approximate group. Then there exists a coset nilprogression ${P}$ of rank and step ${O_K(1)}$ contained in ${A^4}$, such that ${A}$ is covered by ${O_K(1)}$ left-translates of ${P}$ (and hence also by ${O_K(1)}$ right-translates of ${P}$).

Remark 1 Under some mild additional hypotheses (e.g. if the dimensions of ${P}$ are sufficiently large, or if ${P}$ is placed in a certain “normal form”, details of which may be found in this paper), a coset nilprogression ${P}$ of rank and step ${O_K(1)}$ will be an ${O_K(1)}$-approximate group, thus giving a partial converse to Theorem 1. (It is not quite a full converse though, even if one works qualitatively and forgets how the constants depend on ${K}$: if ${A}$ is covered by a bounded number of left- and right-translates ${gP, Pg}$ of ${P}$, one needs the group elements ${g}$ to “approximately normalise” ${P}$ in some sense if one wants to then conclude that ${A}$ is an approximate group.) The mild hypotheses alluded to above can be enforced in the statement of the theorem, but we will not discuss this technicality here, and refer the reader to the above-mentioned paper for details.

By placing the coset nilprogression in a virtually nilpotent group, we have the following corollary in the global case:

Corollary 2 Let ${A}$ be a finite ${K}$-approximate group in an ambient group ${G}$. Then ${A}$ is covered by ${O_K(1)}$ left cosets of a virtually nilpotent subgroup ${G'}$ of ${G}$.

In this final set of notes, we give some applications of the above results. The first application is to replace “${K}$-approximate group” by “sets of bounded doubling”:

Proposition 3 Let ${A}$ be a finite non-empty subset of a (global) group ${G}$ such that ${|A^2| \leq K |A|}$. Then there exists a coset nilprogression ${P}$ of rank and step ${O_K(1)}$ and cardinality ${|P| \gg_K |A|}$ such that ${A}$ can be covered by ${O_K(1)}$ left-translates of ${P}$, and also by ${O_K(1)}$ right-translates of ${P}$.

We will also establish (a strengthening of) a well-known theorem of Gromov on groups of polynomial growth, as promised back in Notes 0, as well as a variant result (of a type known as a “generalised Margulis lemma”) controlling the almost stabilisers of discrete actions of isometries.

The material here is largely drawn from my recent paper with Emmanuel Breuillard and Ben Green.

We now turn to Perelman’s second scale-invariant monotone quantity for Ricci flow, now known as the Perelman reduced volume. We saw in the previous lecture that the monotonicity for Perelman entropy was ultimately derived (after some twists and turns) from the monotonicity of a potential under gradient flow. In this lecture, we will show (at a heuristic level only) how the monotonicity of Perelman’s reduced volume can also be “derived”, in a formal sense, from another source of monotonicity, namely the relative Bishop-Gromov inequality in comparison geometry (which has already been alluded to in previous lectures). Interestingly, in order to obtain this connection, one must first reinterpret parabolic flows such as Ricci flow as the limit of a certain high-dimensional Riemannian manifold as the dimension becomes infinite; this is part of a more general philosophy that parabolic theory is in some sense an infinite-dimensional limit of elliptic theory. Our treatment here is a (liberally reinterpreted) version of Section 6 of Perelman’s paper.

In the next few lectures we shall give a rigorous proof of this monotonicity, without using the infinite-dimensional limit and instead using results related to the Li-Yau-Hamilton Harnack inequality. (There are several other approaches to understanding Perelman’s reduced volume, such as Lott’s formulation based on optimal transport, but we will restrict attention in this course to the methods that are in Perelman’s original paper.)