Kathryn Mann

Papers and preprints

1. Topological stability of relatively hyperbolic groups acting on their boundaries . With Jason Manning and Teddy Weisman.
   We show that a sufficiently small perturbation of the standard boundary action, if assumed on each parabolic subgroup to be a perturbation by semi-conjugacy, is in fact always globally semi-conjugate to the standard action.
2. Anosov flows with the same periodic orbits. With Thomas Barthelme and Sergio Fenley.
   The paper "Orbit equivalence of pseudo-Ansoov flows" (below) shows that flows are essentially classified by the set of free homotopy classes of their periodic orbits. This paper makes the notion of "essentially" precise, describing exactly what features and constructions lead to Anosov flows with the same free homotopy data.
3. Orbit equivalences of pseudo-Anosov flows. With Thomas Barthelme and Steven Frankel.
   We show that, in most cases, the transitive pesudo-Anosov flows on a closed 3-manifold are completely classified by the algebraic data of the set of free homotopy classes of their periodic orbits. (In general, we give a complete invariant of such flows up to orbit equivalence; in special cases this requires some sign choices in addition to the free homotopy class data above, depending on the topology of the orbit space).
4. Stability of hyperbolic groups acting on their boundaries. With Jason Manning and Teddy Weisman.
   We show that any action of a discrete Gromov hyperbolic group on its boundary is stable in the sense of topological dynamics: small perturbations contain the original action as topological factors.
5. Groups acting at infinity. Survey article.
   A survey paper around rigidity theorems that arise from looking to infinity, to appear in Proceedings of the ICM.
6. The action of the (2,3,7) homology group on its space of left orders. With Michelle Triestino.
   A short note giving an example of a finitely generated group where the conjugation action on its space of left orders has two minimal components.
7. Two results on end spaces of infinite type surfaces. With Kasra Rafi.
   We answer two questions about the topology of end spaces of infinite type surfaces: existence of surfaces with non-self similar end space but a unique maximal end, and a proof that the partial order defined previously by the authors identifies ends up to local homeomorphism.
8. Stability for hyperbolic groups acting on boundary spheres. With Jason Manning.
   We show that the action of any Gromov-hyperbolic group with sphere boundary on this sphere is stable in the sense of topological dynamics: nearby actions have the standard action as a topological factor.
9. Rotation sets and actions on curves. With Jonathan Bowden, Sebastian Hensel, Emmanuel Militon, and Richard Webb.
   Advances in Mathematics 408 B, Oct. 2022.
   We study the action of homeomorphisms of surfaces on the fine curve graph (of Bowden-Hensel-Webb) and the relationship with the classical rotation set for torus homeomrophisms.
10. Orbit equivalences of R-covered Anosov flows and applications. With Thomas Barthelme.
    A spectral rigidity theorem for R-covered Anosov flows on 3-manifolds, with applications to contact geometry. Appendix with Jonathan Bowden.
    To appear in Geometry and Topology.
11. (Expository) The structure of homeomorphism and diffeomorphism groups.
    A colloquium-style article written for a general mathematical audience. Notices of the AMS, vol 68 no. 4, (April 2021). Published version here.
12. On the bordism group for group actions on the torus. With Sam Nariman.
    Annales de l'Institut Fourier, Volume 72 (2022) no. 3, pp. 989-1009.
    We study the bordism problem for group actions on the torus and give examples of groups acting on the torus by diffeomorphisms isotopic to the identity that cannot be extended to an action on a bounding 3-manifold
13. There are no exotic actions of diffeomorphism groups on 1-manifolds. With Lei Chen.
    We give a short and self contained proof of the following. For all cases where Diff^r_c(M) is known to be simple the existence of a nontrivial action of such a group on a 1-manifold by homeomorphisms implies M is one-dimensional and the action induced by embeddings of M. This solves a conjecture of Matsumoto.
    To appear in Groups Geometry and Dynamics
14. Large scale geometry of big mapping class groups. With Kasra Rafi.
    We use Rosendal's theory of coarse geometry for non-locally compact groups to study the quasi-isometry types of the mapping class groups of infinite type surfaces.
    In Geometry and Topology. The version linked here has two typos corrected.
15. C^0 stability of boundary actions and inequivalent Anosov flows. With Jonathan Bowden.
    Annales Scientifique de l'ENS. 4e serie, t. 55, 2022. p.1003-1046.
    We prove C^0 structural stability for the action of the fundamental group of a negatively curved manifold on the boundary at infinity of its universal cover.
    Using related techniques, we give a new proof of global rigidity of geometric surface group actions on S^1, and answer a problem from Kirby's problem list:
    For any N, there is a compact hyberbolic 3-manifold supporting N nonconjugate skew-Anosov flows.
16. Reconstructing maps out of groups. With Maxime Wolff.
    When is the algebraic structure of a group of homeomorphisms enough to reconstruct the action of a given homeomorphism?
    We answer this question and give applications to "critical regularity": groups of C^r diffeomorphisms of 1-manifolds that don't admit any actions of higher regularity.
    To appear in Annales Scientifiques de l'ENS.
17. Dynamical and topological obstructions to extending group actions. With Sam Nariman.
    Mathematische Annalen, 377.3 (2020), 1313-1338.
    We define and discuss the bordism group of a group action, and give obstructions to extending a group action on a surface to an action on a 3-manifold bounded by that surface.
18. Automatic continuity for homeomorphism groups of some noncompact manifolds.
    A short note proving automatic continuity in some additional settings.
    This includes homeomorphism groups of manifodls homeomorphic to the interior of a compact manifold with boundary, and mapping class groups of manifolds with a cantor set of ends.
    To appear in Michigan Math J.
19. Structure theorems for actions of homeomorphism groups. With Lei Chen.
    We give an orbit classification theorem and a general structure theorem for actions of groups of homeomorphisms and diffeomorphisms on manifolds. Applications include solutions to multiple problems in a program initiated by Ghys.
20. Rigidity of mapping class group actions on S^1. With Maxime Wolff.
    Geometry and Topology, Volume 24, Number 3 (2020), 1211-1223
    We show that every action of the group of automorphisms of a surface group on the circle is either conjugate to the standard action on the (Gromov) boundary of the group, or factors through a finite group.
    To appear in Geometry and Topology.
21. Realization problems for diffeomorphism groups With Bena Tshishiku.
    A survey/problems list on Nielsen realization problems and their friends. In the Proceedings of the Georgia International Topology festival.
22. A characterization of Fuchsian actions by topological rigidity. With Maxime Wolff.
    Pacific Journal of Math 302.1 (2019) 181-200
    We prove a converse to a rigidity theorem of Matsumoto. This work can be read as an introduction to some of the ideas in the paper Rigidity and geometricity below.
23. Rigidity and geometricity for surface group actions on the circle. With Maxime Wolff.
    We show that the only source of strong topological rigidity for surface group actions on the circle is an underlying geometric structure.
    This is the converse to the main result in my paper "Spaces of surface group representations."
24. Unboundedness of some higher Euler classes.
    Algebraic and Geometric Topology 20 (2020) 1221-1234
    The Milnor-Wood inequality is the statement that the Euler class for flat, topological circle bundles is bounded; this paper shows that analogous classes for flat Seifert fibered 3-manifold bundles are not.
25. Ping-pong configurations and circular orders on free groups. With Dominique Malicet, Cristobal Rivas, and Michele Triestino.
    Groups, Geometry, Dynamics 13.4 (2019), 1195-1218.
    This paper describes isolated circular orders on free groups, answering a question from previous work with Rivas.
26. On the number of circular orders on a group. With Adam Clay and Cristobal Rivas.
    Journal of Algebra 504 (2018) 336-363.
    We classify the groups with finitely many circular orders and give applications to left-orderable groups.
27. Strong distortion in transformation groups. With Frederic Le Roux.
    Bulletin of the London Math. Soc. 50.1 (2018), 46-62.
28. Group orderings, dynamics, and rigidity. With Cristobal Rivas.
    Annales de l'Institut Fourier Volume 68.4 (2018), 1399-1445,
29. The large-scale geometry of homeomorphism groups. With Christian Rosendal.
    Ergodic theory and Dynamical Systems 38.7 (2018), 2748-2779.
30. PL(M) has no Polish group topology.
    Fundamenta Mathematicae 238 (2017), 285-296.
31. Rigidity and flexibility of group actions on S^1.
    In the Handbook of group actions vol 4. L. Ji, A. Papadopoulos, and S.-T. Yau, eds, 2018
32. Automatic continuity for homeomorphism groups and applications.
    With an appendix on the structure of groups of germs of homeomorphism, written with Frederic Le Roux.
    Geometry & Topology 20-5 (2016), 3033-3056.
33. A short proof that the group of compactly supported diffeomorphisms on a manifold is perfect
    following a strategy of Haller, Rybicki and Teichmann. In New York J. Math 22 (2016), 49-55.
34. Left-orderable groups that don't act on the line.
    Math. Zeit. 280 no 3 (2015) 905-918
35. Spaces of surface group representations.
    Inventiones Mathematicae. 201, Issue 2 (2015), 669-710. (link to published version)
36. Diffeomorphism groups of balls and spheres.
    New York J. Math. 19 (2013) 583-596.
37. The simple loop conjecture is false for PSL(2,R).
    Pacific Journal of Mathematics 269-2 (2014), 425-432.
38. Homomorphisms between diffeomorphism groups.
    Ergodic Theory and Dynamical Systems, 35 no. 01 (2015) 192-214.
39. Bounded orbits and global fixed points for groups acting on the plane.
    Algebraic and Geometric Topology 12 (2012) 421-433
40. My dissertation, Components of representation spaces (2014) mostly overlaps with the content of the paper "Spaces of surface group representations" above, although I also very briefly discussed rigidity of universal circle actions of 3-manifold groups, and the thurston norm, at the end.

• Some advertisement of my recent work in the AMS sectional sampler for the 2018 fall meeting.
• Illumination and Security with Alex Wright.
  An exposition of some problems involving translation surfaces, from MSRI's dynamics on moduli spaces program.

Lecture series:

1. Lectures on homeomorphism and diffeomorphism groups (notes from 2015 summer school, not highly polished ~40 pages)
   Related: many lecture notes from a seminar on Cohomology of diffeomorphism groups here .
2. Do-it-yourself Hyperbolic Geometry. A course I taught at Mathcamp.
       Notes are a work in progress, feedback welcome!
3. The mini-course I taught at "Beyond Uniform Hyperbolicity 2015" turned into the survey paper Rigidity and flexibility of group actions on S^1.