# Workshop on classifying spaces of finite groups of Lie type

## University of Copenhagen, July 8–12, 2019

This informal workshop will focus on the interactions between the cohomology, homotopy, and $p$-local theory of finite groups of Lie type, $p$-compact groups, and the free loop spaces of classifying spaces of $p$-compact groups.

Topics for investigation include:

• String topology structures on the cohomology of finite groups of Lie type, along the lines of the preprint String topology of finite groups of Lie typeCan the "fundamental class" of that paper be understood as part of a duality phenomenon? Can one prove the general Tezuka conjecture stating that $H^*(G(q);\mathbb{F}_\ell)$ and $H^*(LBG;\mathbb{F}_\ell)$  are isomorphic when $q$ is congruent to 1 mod $\ell$, and its twisted generalisation for arbitrary $q$?
• Cyclotomic root systems and root data. What is the precise relationship between the story aimed at spetses, developed e.g. in the paper by Broue, Corran and Michel, and the story for $p$-compact groups, described e.g. in the paper by Andersen and Grodal on the classification of 2-compact groups? Is there a classification of cyclotomic root data, i.e. root data over $\mathbb{Z}[\zeta_d]$ (not just root systems)? Can part of the $p$-compact theory be lifted to over $\mathbb{Z}[\zeta_d]$?

• How the above questions relate to classical questions about subgroups, cohomology, and Adams operations on these groups.

• Other similar topics, following input from participants!

### Program

The workshop will consist of approximately 2 hours of background lecture series each morning supplemented by an individual research talk in the afternoon.  The background lectures will include:

• Jean Michel: Cyclotomic root systems and $p$-local structure of finite groups of Lie type
• Jesper Grodal: $p$-compact groups and their fixed point subgroups
• Jason Semeraro: Spetses and counting conjectures
• Anssi Lahtinen: String topology of finite groups of Lie type.

### Schedule (subject to change)

All talks will be in Auditorium 8. There will be cake and fruit in the lunch room on the 4th floor of the math department daily at 14:45.

Monday:

9:30 Jean Michel: Cyclotomic root data (with focus on the parts which may relate to p-compact groups and spetses).

11:00 Jason Semeraro: Counting conjectures. (Both background material, and material from arXiv:1810.01453.)

15:15 Bob Oliver: Automorphisms of fusion and linking systems of finite simple groups.

Tuesday:

9:30 Jesper Grodal: Introduction to p-compact groups and homotopy finite groups of Lie type.

11:00 Jean Michel: Introduction to $\Phi_d$ structure in finite groups of Lie type.

15:15 Bob Oliver: Tameness for fusion systems of finite groups of Lie type.

Wednesday:

9:30 Jesper Grodal: Introduction to p-compact groups and homotopy finite groups of Lie type (continued).

11:00 Jason Semeraro: Spetses and p-compact groups. (Background material on the idea of spetses from papers of Broue-Malle-Michel; see e.g. Malle's ICM talk, and arXiv:1906.00898.)

13:30 Jean Michel et al: More on spetses; Q&A following up on Jason's talk.

15:15 Jesper Grodal: Introduction to p-compact groups and homotopy finite groups of Lie type (continued).

Thursday:

9:30 Jesper Michael Møller: From homotopy Lie groups to homotopy Chevalley groups -- and back again!.

11:00 Anssi Lahtinen: String topology of finite groups of Lie type I.

13:30 Jean Michel

Friday:

9:30 Anssi Lahtinen: String topology of finite groups of Lie type II.

11:00: What we have learned / TBD.

### Registration

To register for the workshop, please email Anssi Lahtinen at lahtinen@math.ku.dk.