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
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 1cohomology, in the sense that if $H$ is any
finite group and $V$ is any faithful, absolutely irreducible Hmodule, 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 1cohomology 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: