# Cohomology bounds and growth rates

June 4 to June 8, 2012

at the

American Institute of Mathematics, Palo Alto, California

organized by

Robert Guralnick, Terrell Hodge, Brian Parshall, and Leonard Scott

This workshop, sponsored by AIM and the NSF, will be devoted to questions associated to the following 1984 conjecture of Guralnick's: There exists a ''universal constant'' $C$ which bounds 1-cohomology, in the sense that if $H$ is any finite group and $V$ is any faithful, absolutely irreducible H-module, then $$\dim H^1(H, V ) \le C.$$

Over 25 years later, this conjecture remains open, but some recent developments have revealed new avenues for investigation. For example, relaxing the constraint on the term ''universal'' a bit, it has been recently shown that in the case of finite groups of Lie type $H = G(q)$, there are constants $C(\Phi)$, depending only on the root system $\Phi$ of the associated algebraic group $G$, which bound $\dim H^1(H, V )$, the 1-cohomology as above.

Significant topics envisioned for workshop investigations include:

• The original conjecture's status and intermediate progress.
• In the case of finite groups of Lie type $H = G(q)$, even if the original conjecture should fail, explore the growth rates for the constants $C(\Phi)$, that is, study possible growth with respect to the Lie rank in the simple groups case.
• Related, rich ''growth'' questions. For example, for a fixed $n$, $\max_L \dim H^n (G, L)$ is finite, as $G$ varies over all semisimple algebraic groups with root system $\Phi$ in any positive characteristic, and $L$ is an irreducible rational $G$-module. Consider the growth rate of the sequence $\{\max_L \dim H^n(G, L)\}$.
• Formulate and investigate parallel ''growth'' theories for the finite groups case.
• Consider other related questions for higher degree cohomology for finite and algebraic groups, utilizing interrelationships between finite and algebraic groups (and quantum groups); progress towards a better ''generic cohomology'' theory in higher cohomological degrees would be ideal.
• Consequences and related applications of the conjecture and related homological growth questions, such as to maximal subgroups of finite groups, questions about generators and relations and other computational group theory issues, and more.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.