SFB 878:  Groups, Geometry and Actions

• Subject Area: Mathematics
• Term: from 2010 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 129719356

• Final Report Year: 2019

Final Report Abstract

The CRC 878 ”Groups, Geometry and Actions” applied a uniform approach for studying a wide range of mathematical topics. We will now describe some of the most notable results obtained in 2018/2–2019/1. The research areas were A. Algebra and logic, B. Geometry and topology, C. Noncommutative geometry and probability theory. A. In the modular representation theory of p-adic groups a differential graded Hecke algebra was constructed whose derived category is equivalent to the derived category of smooth group representations. Moduli spaces of global G-shtukas were constructed for arbitrary flat affine group schemes G of finite type. The analogue of the Langlands-Rapoport conjecture for global G-shtukas was proved. The Rapoport-Zink conjecture concerning the image of the period morphism for p-divisible groups was established. The universal family of semistable p-adic Galois representations was constructed. A p-adic analogue of the Uhlenbeck-Yau theory of semistable vector bundles was established. For the first time natural dynamical systems attached to arithmetic schemes were constructed whose period orbits correspond to the closed points. This had been an open problem even in the simplest case. In the representation theory of algebras, tilting modules were classified by combinatorical data and polytopes. The Hodge theory of vanishing cycles was used for a partial proof of the quantum cluster positivity conjecture. Natural examples of ample stable theories were constructed. The first examples of sharply 2- and 3-transitive groups not containing a nontrivial abelian normal subgroup were found. The existence of such groups had been an open question for decades. B. In differential geometry the phenomena were clarified that occur when passing from the smooth to the singular space. Using the Ricci flow the classification of non-collapsed manifolds with almost non-negative curvature operator was achieved. It was also shown that immortal homogeneous Ricci flow solutions subconverge to a non-gradient Ricci soliton. The Farrell-Jones conjecture was proved for many classes of groups. E.g. CAT (0)-groups, cocompact lattices in Lie groups, GLn (Z), relative hyperbolic groups, mapping class groups of surfaces. Using index theory and cobordism theory it was shown that the global structure of the space R+ (M) of Riemannian metrics of positive scalar curvature on high-dimensional manifolds has a very rich topology. In particular, R+ (Sd) has the structure of an infinite loop space. A classical question about the Pontrjagin classes of topological bundles could be answered: The higher Pontrjagin classes of nonlinear Rn-bundles do not vanish, in contrast to the linear situation. C. New classes of super expanders were found. These graphs are very useful and notoriously difficult to construct. The Rosenberg conjecture was solved: A discrete group is amenable if and only if its reduced group-C*-algebra is quasidiagonal. This result was the final ingredient for the classification of simple, separable, unital, UCT C*-algebras with finite nuclear dimension. The K-theory of the semigroup C*-algebras of ax + b-semigroups for rings of integers of a number field was calculated. The two-point function of the φ4 -QFT model on a noncommutative geometry was calculated explicitely. Previously, this had been achieved only in very rare cases. Limits of random recursive structures were determined using stochastic fixpoint equations. This led to a complete description of all solutions of the equations and to an analysis of leader-election procedures and their connection to coalescence processes. In the theory of random matrices with correlated entries a phase transition was discovered. In the case of weak correlation Wigner’s semicircle law still holds but strong correlation of the entries on the diagonal leads to free convolutions of this law with some other distributions known from random matrix theory.

Publications

• Period spaces for Hodge structures in equal characteristic. Annals of Math., 173:1241–1358, 2011
  U. Hartl
  
• The topology of a semisimple Lie group is essentially unique. Adv. Math., 228(5):2623–2633, 2011
  Linus Kramer
  
• The Borel conjecture for hyperbolic and CATp0q-groups. Ann. of Math. (2), 175(2):631–689, 2012
  Arthur Bartels and Wolfgang Lück
  
• A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities. J. Reine Angew. Math., 679:223–247, 2013
  Burkhard Wilking
  
• A phase transition for the limiting spectral density of random matrices. Electron. J. Probab., 18:no. 17, 17, 2013
  Olga Friesen and Matthias Löwe
  
• C*-algebras of Toeplitz type associated with algebraic number fields. Math. Ann., 355(4):1383–1423, 2013
  J. Cuntz, C. Deninger, and M. Laca
  
• On a conjecture of Rapoport and Zink. Invent. Math., 193:627–696, 2013
  U. Hartl
  
• SK1 and Lie algebras. Math. Annalen, 357:1455–1483, 2013
  P. Schneider and O. Venjakob
  
• Coarse equivalences of Euclidean buildings. Adv. Math., 253:1–49, 2014. With an appendix by Jeroen Schillewaert and Koen Struyve
  Linus Kramer and Richard M. Weiss
  
• K- and L-theory of group rings over GLn (Z). Publ. Math. Inst. Hautes Etudes Sci., 119:97–125, 2014
  Arthur Bartels, Wolfgang Lück, Holger Reich, and Henrik Rüping
  
• Pro-p-Iwahori-Hecke algebras are Gorenstein. J. Inst. Math. Jussieu, 13:753–809, 2014
  R. Ollivier and P. Schneider
  
• Self-dual noncommutative ϕ4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory. Comm. Math. Phys., 329(3):1069–1130, 2014
  Harald Grosse and Raimar Wulkenhaar
  
• The Farrell-Jones conjecture for cocompact lattices in virtually connected Lie groups. J. Amer. Math. Soc., 27(2):339–388, 2014
  A. Bartels, F. T. Farrell, and W. Lück
  
• Nuclear dimension and Z-stability. Invent. Math., 202:893–921, 2015
  Y. Sato, S. White, and W. Winter
  
• Purity for graded potentials and quantum cluster positivity. Compositio Mathematica, 151:1913–1744, 2015
  B. Davison, D. Maulik, J. Schürmann, and B. Szendroi
  
• Transitive actions of locally compact groups on locally contractible spaces. J. Reine Angew. Math., 702:227–243, 2015
  Karl H. Hofmann and Linus Kramer
  
• Sharply 3-transitive groups. Adv. Math., 286:722–728, 2016
  Katrin Tent
  
• Simplicity of the automorphism groups of some Hrushovski constructions. Ann. Pure Appl. Logic, 167(1):22–48, 2016
  David M. Evans, Zaniar Ghadernezhad, and Katrin Tent
  
• Surgery stable curvature conditions. Math. Ann., 365(1-2):13–47, 2016
  Sebastian Hoelzel
  
• A leader-election procedure using records. Ann. Probab., 45(6B):4348–4388, 2017
  Gerold Alsmeyer, Zakhar Kabluchko, and Alexander Marynych
  
• Characteristic classes of symmetric products of complex quasi-projective varieties. J. Reine Angew. Math., 728:35– 63, 2017
  S. Cappell, L. Maxim, J. Schürmann, J. Shaneson, and S. Yokura
  
• Coarse flow spaces for relatively hyperbolic groups. Compos. Math., 153(4):745– 779, 2017
  A. Bartels
  
• Convex hulls of random walks, hyperplane arrangements, and Weyl chambers. Geom. Funct. Anal., 27(4):880–918, 2017
  Zakhar Kabluchko, Vladislav Vysotsky, and Dmitry Zaporozhets
  
• Convex hulls of random walks: expected number of faces and face probabilities. Adv. Math., 320:595–629, 2017
  Zakhar Kabluchko, Vladislav Vysotsky, and Dmitry Zaporozhets
  
• Exact solution of matricial Φ3 2 quantum field theory. Nuclear Phys. B, 925:319–347, 2017
  Harald Grosse, Akifumi Sako, and Raimar Wulkenhaar
  
• Infinite loop spaces and positive scalar curvature. Invent. Math., 209(3):749–835, 2017
  Boris Botvinnik, Johannes Ebert, and Oscar Randal-Williams
  
• Nonextensive condensation in reinforced branching processes. Ann. Appl. Probab., 27(4):2539–2568, 2017
  S. Dereich, C. Mailler, and P. Mörters
  
• On the Berger conjecture for manifolds all of whose geodesics are closed. Invent. Math., 210(3):911–962, 2017
  Marco Radeschi and Burkhard Wilking
  
• Parallel transport for vector bundles on p-adic varieties
  C. Deninger and A. Werner
  
• Quasidiagonality of nuclear C*-algebras. Ann. of Math. (2), 185:229–284, 2017
  A. Tikuisis, S. White, and W. Winter
  
• Exotic crossed products and the Baum-Connes conjecture. J. Reine Angew. Math., 740:111–159, 2018
  Alcides Buss, Siegfried Echterhoff, and Rufus Willett
  
• Immortal homogeneous Ricci flows. Invent. Math., 212(2):461–529, 2018
  Christoph Böhm and Ramiro A. Lafuente
  
• Nonarchimedean bornologies, cyclic homology and rigid cohomology. Doc. Math., 23:1197–1245, 2018
  Guillermo Cortiñas, Joachim Cuntz, Ralf Meyer, and Georg Tamme
  
• Revisiting homogeneous spaces with positive curvature. J. Reine Angew. Math., 738:313–328, 2018
  Burkhard Wilking and Wolfgang Ziller
  
• Building-like geometries of finite Morley rank. J. Eur. Math. Soc. (JEMS), 21(12):3739–3757, 2019
  Isabel Müller and Katrin Tent
  
• Index theory in spaces of manifolds. Mathematische Annalen, 374(1–2):931–962, June 2019
  J. Ebert
  
• Superexpanders from group actions on compact manifolds. Geom. Dedicata, 200(1):287–302, 2019
  Tim de Laat and Federico Vigolo
  
• The Farrell-Jones conjecture for mapping class groups. Invent. Math., 215(2):651–712, 2019
  Arthur Bartels and Mladen Bestvina
  
• The ricci flow under almost non-negative curvature conditions. Invent. Math., 217(1):95–126, 2019
  Richard Bamler, Esther Cabezas-Rivas, and Burkhard Wilking