Justine Falque (Université Paris-Sud)

Classification of oligomorphic groups with polynomial profiles, conjectures of Cameron and Macpherson.

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Let $G$ be a group of permutations of a denumerable set $E$. The profile of $G$ is the function $f$ which counts, for each $n$, the (possibly infinite) number $f(n)$ of orbits of $G$ acting on the $n$-subsets of $E$. When $f$ takes only finite values, $G$ is called oligomorphic.

Counting functions arising this way, and their associated generating series, form a rich yet apparently strongly constrained class. In particular, Cameron conjectured in the late seventies that, whenever the profile $f(n)$ is bounded by a polynomial (we say that $G$ is $P$-oligomorphic), it is asymptotically equivalent to a polynomial. In 1985, Macpherson further asked whether the orbit algebra of $G$ (a graded commutative algebra invented by Cameron and whose Hilbert function is $f$) was finitely generated.

After providing some context and definitions of the involved objects, this talk will outline the proof of a classification result of all (closed) $P$-oligomorphic groups, of which the conjectures of Cameron and Macpherson are corollaries.

The proof exploits classical notions from group theory (notably block systems and their lattice properties), commutative algebra, and invariant theory. This research was a joint work with Nicolas Thiéry.