Periodic Reporting for period 4 - GROGandGIN (Growth in Groups and Graph Isomorphism Now)
Periodo di rendicontazione: 2022-02-01 al 2023-07-31
In recent years there has been spectacular progress in studying growth in finite groups. The main issue addressed in the project is to significantly further this progress.
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.
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.
The research group (with various coauthors) worked on several sub-projects of the proposed activity.
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. Motivated by this conjecture we formulated and after many years proved a result analogous to the Product Theorem , the so-called Conjugacy Product Theorem , valid for unbounded rank classical simple groups.. Based on this Dona managed to prove an approximate version of the Liebeck-Nikolov-Shalev conjecture for classical simple groups. These papers are being written.
A consequence of the Product Theorem is that for bounded rank simple groups Babai's conjecture holds, that is 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.
The result of Halasi was extended first by our team to non-orthogonal classical groups over fields of odd order and finally by Eberhard (who is a close associate of our team to all non-orthogonal classical groups.
Combined with the breakhrough paper of Eberhard-Jezernik (Inventiones), this implies that Babai's conjecture holds for 3 random elements in non-orthogonal classical groups. This seems to be a rather strong evidence in favour of the conjecture.
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. In fact rather surprisingly the result is true for homeomorphism groups. Our paper has been submitted to a top journal about a year ago.
We have characterised non-growing sets in bounded rank linear groups confirming fully a conjecture of Helfgott-Lindenstrauss. Our paper has been submitted to a strong mathematical journal at the beginning of this year.
Due to covid and some health problems the PI travelled less than expected, But last year he gave an invited colloquium talk in Cambridge about some important results of the team.
Next year in April he will give an invited lecture course on results related to the Conjugacy Product Theorem at an Oberwolfach symposium. Other team members gave several talks at various conferences about their work.
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. Motivated by this conjecture we formulated and after many years proved a result analogous to the Product Theorem , the so-called Conjugacy Product Theorem , valid for unbounded rank classical simple groups.. Based on this Dona managed to prove an approximate version of the Liebeck-Nikolov-Shalev conjecture for classical simple groups. These papers are being written.
A consequence of the Product Theorem is that for bounded rank simple groups Babai's conjecture holds, that is 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.
The result of Halasi was extended first by our team to non-orthogonal classical groups over fields of odd order and finally by Eberhard (who is a close associate of our team to all non-orthogonal classical groups.
Combined with the breakhrough paper of Eberhard-Jezernik (Inventiones), this implies that Babai's conjecture holds for 3 random elements in non-orthogonal classical groups. This seems to be a rather strong evidence in favour of the conjecture.
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. In fact rather surprisingly the result is true for homeomorphism groups. Our paper has been submitted to a top journal about a year ago.
We have characterised non-growing sets in bounded rank linear groups confirming fully a conjecture of Helfgott-Lindenstrauss. Our paper has been submitted to a strong mathematical journal at the beginning of this year.
Due to covid and some health problems the PI travelled less than expected, But last year he gave an invited colloquium talk in Cambridge about some important results of the team.
Next year in April he will give an invited lecture course on results related to the Conjugacy Product Theorem at an Oberwolfach symposium. Other team members gave several talks at various conferences about their work.
One of the main objectives of the project was to prove a version of the Product Theorem, which holds in unbounded rank. We have obtained such a result, the Conjugacy Product Theorem, at least for classical groups. We are working on the remaining case of alternating groups.
This result and its application to the Liebeck-Nikolov-Shalev conjecture is a real breakthrough.
Another breakthrough is the theorem that for 3 random elements of non-orthogonal classical groups Babai's diameter conjecture holds, another unbounded rank result.
Finally being able to show that the Ghys conjecture on finite subgroups of diffeomorphisms holds, moreover that it holds even for homeomorphism groups is certainly a result beyond the state of art.
This result and its application to the Liebeck-Nikolov-Shalev conjecture is a real breakthrough.
Another breakthrough is the theorem that for 3 random elements of non-orthogonal classical groups Babai's diameter conjecture holds, another unbounded rank result.
Finally being able to show that the Ghys conjecture on finite subgroups of diffeomorphisms holds, moreover that it holds even for homeomorphism groups is certainly a result beyond the state of art.