# Cohomology bounds and growth rates

June 4 to June 8, 2012

at the

American Institute of Mathematics, San Jose, California

organized by

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

## Original Announcement

This workshop 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.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Stabilisation of the LHS spectral sequence for algebraic groups
by  Alison E. Parker and David I. Stewart ,  J. Lie Theory 25 (2015), no. 3, 807–813  MR3384997
First cohomology groups for finite groups of Lie type in defining characteristic
by  Alison E. Parker and David I. Stewart ,  Bull. Lond. Math. Soc. 46 (2014), no. 2, 227–238  MR3194742
Extensions for finite Chevalley groups III: Rational and generic cohomology
by  Christopher P. Bendel, Daniel K. Nakano, and Cornelius Pillen,  Adv. Math. 262 (2014), 484-519  MR3228435
Bounding the dimensions of rational cohomology groups
by  Christopher P. Bendel, Brian D. Boe, Christopher M. Drupieski, Daniel K. Nakano, Brian J. Parshall, Cornelius Pillen, and Caroline B. Wright,  Developments and retrospectives in Lie theory, 51-69, Dev. Math. 38, Springer, Cham, 2014  MR3308777