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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