# Achievements in Research Area F

## Geometric automorphism groups, group cohomology, CAT(0)-spaces.

Ursula Hamenstädt made significant progress toward the third goal formulated in the proposal, an understanding of continuous bounded group cohomology. She obtained a characterisation of elementary groups of isometries of proper hyperbolic spaces via continuous bounded cohomology with twisted coefficients, [F:Ham09a]. These results could be generalised to isometry groups of proper rank one CAT(0)-spaces, [F:Ham08], [F:Ham09c]. Continuous bounded cohomology classes with twisted coefficients for the outer automorphism group of a free group were constructed; moreover, bounded cohomology was related to cross ratios and some rigidity results were established, [F:Ham09b].

Lytchak, now a Heisenberg fellow, investigated with Caprace isometry groups of proper CAT(0)-spaces. They characterised groups which fix a point at infinity and extend earlier known rigidity results, [F:CL10]. He also studied geometric properties of isometric actions of compact groups on Riemannian manifolds, their orbifolds and orbit foliations, [F:LT07].

Other geometric results include the proof of biautomaticity of the mapping class group and its quasi-isometric rigidity obtained by Ursula Hamenstädt. Her joint work with Hensel, a comprehensive investigation of the handlebody group, benefited from a special semester at the MSRI in Berkeley (2007) which they attended; Hensel’s visit was financed by HCM. A HCM workshop on subgroups of the mapping class group in 2010, organised by Ursula Hamenstädt, Möller and Zorich, marked an international highlight in the field and initiated research projects.

## Group actions and topological invariants.

The investigators Kreck and Catharina Stroppel (who joined Bonn during the first period) worked on problems related to topologial and algebraic invariants arising from group actions.

Kreck investigated topological properties of manifolds in connection with group actions. In [F:Kre09] he gives the first examples of smooth simply connected closed manifolds which do not admit a nontrivial action of a finite group. In [F:KL09], a complete classification of orientable 4-manifolds with solvable Baumslag-Solitar fundamental group is established. K. Grove (Notre Dame), Bonn Research Chair 2010, worked on the classification of isometric Lie group actions and in particular on circle actions on four manifolds with nonnegative curvature.

Catharina Stroppel constructed in [F:MS09] new knot invariants by categorifying known invariants from quantum groups, providing interesting faithful actions of braid groups via functors on derived categories. Group actions on abelian and derived categories are also used in [F:MS08] to give a complete answer to an old representation theoretic problem of determining Jordan-Hölder multiplicities of induced arbitrary simple modules for semisimple Lie algebras.

## Homogeneous spaces and Kac-Moody groups.

Substantial progress in the direction of the algebraic goals of the proposal was achieved on loop groups, homogeneous spaces and the geometry of affine Grassmannians. Perrin (BJF) exhibited in [F:CMP09] an action of the fundamental group of the automorphism group of any rational homogeneous space on its quantum cohomology. This is done via a realisation of the quantum cohomology as a quotient of the homology of the affine Grassmannian, a homogeneous space under the algebraic loop group of the automorphism group. The cohomology of a homogeneous space under a Kac-Moody group (a generalisation of an algebraic loop group) is studied in [F:CP11] with a combinatorial description of the product of two Schubert classes by explicit formulas for the Littlewood-Richardson coefficient of any (co)minuscule Schubert class.

Pasquier, a postdoc at HCM (’07-’09), studied and classified smooth projective varieties with Picard number one where the automorphism group has two orbits, [F:Pas09]. Perrin obtained moreover results on the quantum cohomology of (co)adjoint varieties, quantum K-theory as well as on the fibres of the Springer resolution and their irreducible and connected components, [F:CP]. These results are related to research projects in Research Area C.

## Affine Deligne-Lusztig varieties and Kisin theory.

Connected components of prominent subvarieties of affine Grassmannians are affine Deligne-Lusztig varieties and Kisin varieties. They were thoroughly investigated in the working group of Rapoport. One of the main results is an affirmative answer to a conjecture of Rapoport that these varieties are equidimensional of predicted dimension. This result was obtained by Eva Viehmann together with a detailed description of the connected components, [F:Vie08], [F:Vie06].

Methods developed for affine Deligne-Lusztig varieties were successfully adapted and generalised to the Kisin theory of deformations of Galois representations using the concept of varieties, so named by Pappas and Rapoport. Caruso and Hellmann [F:Hel09], [F:Hel] obtained interesting results on the structure of Kisin varieties, although the theory of these varieties is still in its infancy. The results of Caruso and Hellmann were obtained under mutual influence due to their interaction during visits of Caruso to Bonn, mostly financed by HCM. One of the highlights was the JTP on Algebra and Number Theory at HIM organised by Eva Viehmann who in turn benefited from a research visit at the University of Chicago which was also financed by HCM.

## Loop groups.

Rapoport, joint with Pappas, defined and investigated loop groups and their associated affine flag varieties in the twisted case [F:PR08b]. This theory was applied to local models of Shimura varieties in the case when the local group is ramified [F:PR09b]. The questions arising here are flatness, the reducedness of the special fiber, as well as the enumeration of the Schubert varieties contributing to the special fiber. The Coherence Conjecture stated in [F:PR08b] and recently proved by X. Zhu is of central importance in the field. The work of Smithling gives explicit realisations of local models for classical groups, and is essential for applications of this theory. Smithling started working in this area during his visit to MPIM, and had lively interactions with Rapoport during his participation in the Junior Trimester Program 2010.