Dear all,

We'd like to announce a "working workshop" (or "research retreat") for women in geometric topology (or geometric group theory) which will take place next summer August 9 -15, 2020 at Rising Wolf ranch in Montana.  We're soliciting applications from women to join our small working groups. Preference will be given to women close to finishing their Ph.D and new postdocs (or at teaching colleges). We hope you would consider applying or/and helping us recruiting potential participants.

Our aim is to have groups of 4-5 women with 1-2 co-leaders per group working on specific projects. This year the potential projects are listed below. Groups may be split into two or postponed depending on the interest from applicants. To apply please fill this form https://math.ou.edu/~jing/wiggd2020/ by February 29, 2020.
  1. Big mapping class groups  (Carolyn Abbott, Priyam Patel): For this workshop, we propose to study subgroups of big mapping class groups, with two broad goals in mind. The first addresses the question of existence of subgroups. In particular, which groups appear as subgroups of big mapping class groups? The second goal is to understand the properties of groups which do arise as subgroups of big mapping class groups. For example, the famous Nielsen realization problem asks whether  finite subgroups of Map(S) can be realized by S, i.e., for G a finite subgroup of Map(S) does there exist a geodesically complete hyperbolic structure on S whose isometry group is G?
  2. From Gromov hyperbolic to CAT(0) groups  (Kim Ruane, Emily Stark): Two possible projects: 1. Boundaries of Gromov hyperbolic groups have rich structure that can be used to distinguish groups up to quasi-isometry. General CAT(0) groups do not have a well defined visual boundary even up to homeomorphism. We will restrict to CAT(0) groups with isolated flats and try to define a quasisymmetric structure similar to the one on hyperbolic groups with the goal to apply it to quasi-isometric rigidity.
    2. The nicest subgroups of a hyperbolic group are its quasi-convex subgroups which are again hyperbolic.  Moreover, the inclusion of a quasi-convex subgroup in a hyperbolic group extends to a continuous map from the boundary of the subgroup to the boundary of the group. Outside of the hyperbolic setting, there is a not an obvious analog for a quasi-convex subgroup. We plan to investigate quasi-convex subgroups in the CAT(0) setting. For subgroups that are not quasi-convex, one may still ask if the inclusion from the subgroup to the group extends to a continuous map between boundaries. Such a map is called a Cannon–Thurston map and there are nice examples for highly distorted free subgroups of hyperbolic hydra groups. We propose to generalize these to CAT(0) groups.
  3. Quasi-isometries (Tullia Dymarz, Natasa Macura): A space may have a large quasi-isometry group that becomes much smaller if you restrict to ``pattern preserving" quasi-isometries. For example you could ask that your quasi-isometry preserves the set of cosets of certain subgroups. This situation shows up for example when studying quasi-isometries of  graphs of groups. We will investigate certain well known spaces and how rigid pattern preserving quasi-isometries can be. Some groups we will consider are SOL and groups related to Baumslag-Solitar groups.
  4. Curve graphs of nonorientable surfaces (Sara Maloni, Jing Tao): Curve graphs of orientable surfaces have been studied and used a lot in many different research directions, but less is known about curve graphs of nonorientable surfaces. These graphs and their dual play an important role also in the study of mapping class group actions on character varieties, systoles, McShane identities, simple length spectra and asymptotic growth of simple closed geodesics. Recently the case of the thrice punctured projective plane N_{1,3} has been studied extensively in very different directions. For the workshop, we want to focus on the case of N_{2,2} and N_{3, 1}, which are, respectively, the connected sum of two projective planes with two punctures, and the connected sum of three projective planes with one puncture, and see if we can generalise some of the results mentioned above. The first step will be to give a concrete description of their curve graphs and their dual graphs, and, after that is accomplished, we will see which other results we can derive.
We hope to form the groups soon after the deadline. We will then send a longer description and a list of background readings for the projects. You should also expect, soon after, a message from your project leaders with more information about how to prepare for the workshop, so that we will use the time at the cabin efficiently to work on such project and make progresses. We hope also to organize some talks/discussions about common questions/issues experienced often in our career, so if there is something you would like to hear about, feel free to let us know as well. Do not worry: we will also have time to experience the beautiful environment where the retreat will be held. Montana and Glacier National Park are awesome at that time of the year. 🙂

If you have any question or doubt, please let us know. We hope to hear from you soon!
Best,
    Jing, Sara and Tullia



--
Tullia Dymarz
Associate Professor
Department of Mathematics
University of Wisconsin, Madison
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