Ph.D

Group : Graphs, ALgorithms and Combinatorics

*Classification of P-oligomorphic groups, conjectures of Cameron and Macpherson*
Starts on

Advisor : THIÉRY, Nicolas

**Funding :**
**Affiliation :** Université Paris-Saclay

**Laboratory :** l'amphithéâtre DIGITÉO du bâtiment CLAUDE SHANNON

**Defended** on 29/11/2019, committee :

Rapporteurs :

Peter Cameron (Queen Mary Univ. of London & Univ. of Saint Andrews)

Pascal Weil (CNRS, Université de Bordeaux)

Jury :

Peter Cameron (Queen Mary Univ. of London & Univ. of Saint Andrews)

Pascal Weil (CNRS, Université de Bordeaux)

Isabelle Guyon (Université Paris Sud)

Maurice Pouzet (Université Claude Bernard Lyon I)

Christophe Tollu (Université Paris-Nord)

Annick Valibouze (Sorbonne Universités)

Nicolas Thiéry (Université Paris Sud)

**Research activities :**
**Abstract :**
Given an infinite permutation group G, consider the function that

maps every natural integer n to the number of orbits of n-subsets,

for the induced action of G on the subsets of elements.

Cameron conjectured that this counting function, the profile of G,

is asymptotically equivalent to a polynomial if it is bounded

above by a polynomial. Another, stronger conjecture was later made

by Macpherson. It involves a certain structure of graded agebra on

the orbits of subsets, created by Cameron, and states that if the

profile of G is bounded by a polynomial, then its orbit algebra is

finitely generated.

The main achievement of the thesis is to classify the permutation

with polynomially bounded profile (up to closure), which in

particular demonstrates the two conjectures. The approach involves

studying the lattice of block systems, an experimental exploration

on computer, and tools from group theory.