T4: Groups and actions

The study of symmetry and space through the medium of groups and their actions has long been a central theme in modern mathematics, indeed one that cuts across a wide spectrum of research within the Cluster. There are two main constellations of activity in the Cluster that coalesce around groups and dynamics as basic objects of study, and these are captured in the three research units collected here.

Units "Groups, dynamics and C*-algebras" and "Entropy, probability and geometry of groups" both focus on aspects of groups and dynamics grounded in measure and topology in their most abstract sense. This research treats infinite discrete groups as geometric or combinatorial objects and employs tools from functional analysis, probability, and combinatorics. The unit "Algebraic groups and Lie groups" examines, in contrast to abstract or discrete groups, groups with additional structure that naturally arise in algebraic and differential geometry.

  • Mathematical fields

    • Model theory and set theory
    • Arithmetic geometry and representation theory
    • Topology
    • Operator algebras and mathematical physics
    • Differential geometry
    • Stochastic analysis

  • Collaborations with other Topics

  • Selected publications and preprints

    since 2019

    $\bullet $ Christoph Böhm and Urs Hartl. Moment map flow on real reductive Lie groups and GIT estimates. arXiv e-prints, June 2024. arXiv:2406.19340.

    $\bullet $ Eusebio Gardella, Shirly Geffen, Julian Kranz, Petr Naryshkin, and Andrea Vaccaro. Tracially amenable actions and purely infinite crossed products. Mathematische Annalen, March 2024. doi:10.1007/s00208-024-02833-9.

    $\bullet $ Simon André and Katrin Tent. Simple sharply 2-transitive groups. Transactions of the American Mathematical Society, 376(06):3965–3993, June 2023. doi:10.1090/tran/8846.

    $\bullet $ Eusebio Gardella, Shirly Geffen, Julian Kranz, and Petr Naryshkin. Classifiability of crossed products by nonamenable groups. J. Reine Angew. Math., 797(0):285–312, April 2023. doi:10.1515/crelle-2023-0012.

    $\bullet $ Konstantin Ardakov and Peter Schneider. Stability in the category of smooth mod-$p$ representations of $\mathrm S \mathrm L_2(\mathbb Q_p)$. arXiv e-prints, April 2023. arXiv:2304.02585.

    $\bullet $ Chiranjib Mukherjee and Konstantin Recke. Haagerup property and group-invariant percolation. arXiv e-prints, March 2023. arXiv:2303.17429.

    $\bullet $ Kang Li, Hung-Chang Liao, and Wilhelm Winter. The diagonal dimension of sub-$\mathrm C^*$-algebras. arXiv e-prints, March 2023. arXiv:2303.16762.

    $\bullet $ Christoph Böhm and Ramiro A. Lafuente. Non-compact Einstein manifolds with symmetry. J. Amer. Math. Soc., February 2023. doi:10.1090/jams/1022.

    $\bullet $ Simon André and Vincent Guirardel. Finitely generated simple sharply 2-transitive groups. arXiv e-prints, December 2022. arXiv:2212.06020.

    $\bullet $ Matthew Emerton, Toby Gee, and Eugen Hellmann. An introduction to the categorical $p$-adic Langlands program. arXiv e-prints, October 2022. arXiv:2210.01404.

    $\bullet $ David Kerr and Hanfeng Li. Entropy, virtual Abelianness, and Shannon orbit equivalence. arXiv e-prints, February 2022. arXiv:2202.10795.

    $\bullet $ David Kerr and Hanfeng Li. Entropy, Shannon orbit equivalence, and sparse connectivity. Math. Ann., 380(3):1497–1562, May 2021. doi:10.1007/s00208-021-02190-x.

    $\bullet $ Jorge Castillejos, Samuel Evington, Aaron Tikuisis, Stuart White, and Wilhelm Winter. Nuclear dimension of simple C$^*$-algebras. Inventiones mathematicae, 224(1):245–290, April 2021. doi:10.1007/s00222-020-01013-1.

    $\bullet $ Oskar Braun, Karl H. Hofmann, and Linus Kramer. Automatic continuity of abstract homomorphisms between locally compact and polish groups. Transform. Groups, 25(1):1–32, March 2020. doi:10.1007/s00031-019-09537-4.

    $\bullet $ Linus Kramer and Olga Varghese. Abstract homomorphisms from locally compact groups to discrete groups. J. Algebra, 538:127–139, November 2019. doi:10.1016/j.jalgebra.2019.07.026.

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  • Recent publications and preprints

    since 2023

    $\bullet $ Christopher Deninger, Theo Grundhöfer, and Linus Kramer. Weyl tensors, strongly regular graphs, multiplicative characters, and a quadratic matrix equation. J. Algebra, 656:170–195, October 2024. doi:10.1016/j.jalgebra.2023.08.028.

    $\bullet $ Philipp Sibbel and Wilhelm Winter. A Cantor spectrum diagonal in O$_2$. arXiv e-prints, September 2024. arXiv:2409.03511.

    $\bullet $ Shirly Geffen and Julian Kranz. Note on C*-algebras associated to boundary actions of hyperbolic 3-manifold groups. arXiv e-prints, July 2024. arXiv:2407.15215.

    $\bullet $ Christoph Böhm and Urs Hartl. Moment map flow on real reductive Lie groups and GIT estimates. arXiv e-prints, June 2024. arXiv:2406.19340.

    $\bullet $ Eusebio Gardella, Shirly Geffen, Rafaela Gesing, Grigoris Kopsacheilis, and Petr Naryshkin. Essential freeness, allostery and Z-stability of crossed products. arXiv e-prints, May 2024. arXiv:2405.04343.

    $\bullet $ Eusebio Gardella, Shirly Geffen, Julian Kranz, Petr Naryshkin, and Andrea Vaccaro. Tracially amenable actions and purely infinite crossed products. Mathematische Annalen, March 2024. doi:10.1007/s00208-024-02833-9.

    $\bullet $ Eusebio Gardella, Shirly Geffen, Petr Naryshkin, and Andrea Vaccaro. Dynamical comparison and Z-stability for crossed products of simple $\rm C^*$-algebras. Advances in Mathematics, 438:109471, February 2024. doi:10.1016/j.aim.2023.109471.

    $\bullet $ Marco Amelio, Simon André, and Katrin Tent. Non-split sharply 2-transitive groups of odd positive characteristic. arXiv e-prints, December 2023. arXiv:2312.16992.

    $\bullet $ Kristin Courtney and Wilhelm Winter. Images of order zero maps. arXiv e-prints, December 2023. arXiv:2312.08215.

    $\bullet $ Corina Ciobotaru, Linus Kramer, and Petra Schwer. Polyhedral compactifications, I. Advances in Geometry, 23(3):413–436, August 2023. doi:10.1515/advgeom-2023-0018.

    $\bullet $ Simon André and Katrin Tent. Simple sharply 2-transitive groups. Transactions of the American Mathematical Society, 376(06):3965–3993, June 2023. doi:10.1090/tran/8846.

    $\bullet $ Christopher Deninger and Michael Wibmer. On the proalgebraic fundamental group of topological spaces and amalgamated products of affine group schemes. arXiv e-prints, June 2023. arXiv:2306.03296.

    $\bullet $ Eugen Hellmann. On the derived category of the Iwahori–Hecke algebra. Compositio Mathematica, 159(5):1042–1110, May 2023. doi:10.1112/S0010437X23007145.

    $\bullet $ Eusebio Gardella, Shirly Geffen, Julian Kranz, and Petr Naryshkin. Classifiability of crossed products by nonamenable groups. J. Reine Angew. Math., 797(0):285–312, April 2023. doi:10.1515/crelle-2023-0012.

    $\bullet $ Shirly Geffen and Dan Ursu. Simplicity of crossed products by FC-hypercentral groups. arXiv e-prints, April 2023. arXiv:2304.07852.

    $\bullet $ Konstantin Ardakov and Peter Schneider. Stability in the category of smooth mod-$p$ representations of $\mathrm S \mathrm L_2(\mathbb Q_p)$. arXiv e-prints, April 2023. arXiv:2304.02585.

    $\bullet $ Kristin Courtney and Wilhelm Winter. Nuclearity and CPC$^*$-systems. arXiv e-prints, April 2023. arXiv:2304.01332.

    $\bullet $ Chiranjib Mukherjee and Konstantin Recke. Haagerup property and group-invariant percolation. arXiv e-prints, March 2023. arXiv:2303.17429.

    $\bullet $ Kang Li, Hung-Chang Liao, and Wilhelm Winter. The diagonal dimension of sub-$\mathrm C^*$-algebras. arXiv e-prints, March 2023. arXiv:2303.16762.

    $\bullet $ Michael Wiemeler. On circle actions with exactly three fixed points. arXiv e-prints, March 2023. arXiv:2303.15396.

    $\bullet $ Christoph Böhm and Ramiro A. Lafuente. Non-compact Einstein manifolds with symmetry. J. Amer. Math. Soc., February 2023. doi:10.1090/jams/1022.

    $\bullet $ Linus Kramer and Markus J. Stroppel. Hodge operators and exceptional isomorphisms between unitary groups. Journal of Lie Theory, 33(1):329–360, January 2023. URL: www.heldermann.de/JLT/JLT33/JLT331/jlt33015.htm.

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    further publications

Back to research programme

Groups, dynamics and C*-algebras

© MM/vl

Investigators: Geffen, Kerr, Tent, Winter

In this unit the connection to operator algebras is fundamental, influencing many other mathematical areas, including the rigidity theory of Lie groups and their lattices, and the study of manifolds and geometry through invariants like K-theory. Despite the diversity of problems and research paths, a common philosophy unites them: unravelling structural relationships at a fundamental level within a broad landscape shaped by key concepts such as finite approximation and amenability, freeness and tree-like/hyperbolic geometry and property-(T)-style rigidity.

Entropy, probability and geometry of groups

© MM/vl

Investigators: Deninger, Kerr, Mukherjee

This unit circles around three basic themes within the expanded field of group-geometric ergodic theory and probability: (i) the application of tools from Ornstein theory and geometric group theory to advance a general program around orbit equivalence and entropy, with a special focus on rigidity problems for Bernoulli actions, (ii) the development of noncommutative cyclotomy conditions that will determine when an algebraic action an amenable or sofic group has zero entropy, and (iii) the use of invariant percolation to investigate geometric and analytic phenomena in infinite groups such as property (T) and the Haagerup property, in particular through the operator-algebraic lens of Roe algebras.

Algebraic groups and Lie groups

© MM/vl

Investigators: Böhm, Hartl, Hellmann, Kramer, Lourenço, Schneider, Viehmann

This unit examines, in contrast to abstract or discrete groups, groups with additional structure that naturally arise in algebraic and differential geometry: algebraic groups and Lie groups. One of the research directions here focusses on algebraic groups and their actions that arise in the context of the Langlands programme, where representations of $p$-adic Lie groups are one of the main objects of study. A second point of focus includes more general locally compact groups and their associated (Bruhat–Tits) buildings and actions of real reductive groups in the context of Riemannian geometry.