Periodic Reporting for period 2 - GROGandGIN (Growth in Groups and Graph Isomorphism Now)
Reporting period: 2019-02-01 to 2020-07-31
A central result in this new area obtained independently by Pyber-Szabó and Breuillard-Green-Tao is the Product Theorem for finite simple groups of ""bounded dimension"". This result has started a number of deep developments throughout mathematics which, we believe, are important for the society.
For example it has deep consequences for diameter bounds and expansion of Cayley graphs of simple groups, a much studied subject. These consequences are in turn key tools in theory of the Affine Sieve of Bourgain-Gamburd-Sarnak. Further developments include an analogous Product Theorem in simple Lie groups which led to a spectral gap theorem, results in the theory of random walks in Euclidean spaces by Lindenstrauss and Varjú, and more.
The overall objectives include obtaining generalisations of the Product Theorem and its consequences
with particular attention on results in which the the dependence on the dimension/rank is removed."
Some already resulted in accepted papers, some are at the preprint stage and some are currently being written.
Our basic strategy towards obtaining unbounded dimension results is to consider some difficult conjectures which in the bounded rank case have been proved as consequences of the Product Theorem.
One of these conjectures, due to Liebeck-Nikolov-Shalev concerns decompositions of simple groups as products of conjugates of a subset. The PI, Gill and Szabó further generalised this conjecture to conjugates of different subsets and confirmed the generalisation in the bounded rank case using the Product Theorem. The PI and Maróti have also confirmed the new conjecture for conjugacy classes, another important special case. Their result generalises one of the deepest known results about unbounded rank simple groups, the famous diameter bound due to Liebeck-Shalev.
Another famous conjecture , due to Babai predicts that connected Cayley graphs of finite simple groups
have a polylogarithmic diameter. The main open, previously almost untouched case is that of unbounded rank special linear groups over finite fields. In a most welcome development Halasi (a member of the group) confirmed the conjecture for prime fields when the generating set contains a transvection. On the other hand the PI, Halasi, Maróti and Qiao have greatly improved earlier diameter bounds for all classical simple groups.
Another sub-project is the ""Jordan-type"" conjecture of Ghys for finite groups G acting by diffeomorphisms on a compact manifold M. Confirming this conjecture the PI,Csikós and Szabó proved that such a group G has a nilpotent subgroup of index bounded in terms of M."
The (longish) proof of the Ghys conjecture certainly represents progress beyond the state of art.
It involves a combination of developments in finite group theory and in the theory of manifolds.