One-day Workshop on Dynamics, Geometry and Groups

Venue: AORC seminar room, 3rd floor, General Studies building (building 51), Sungkyunkwan university, Suwon, Korea

Invited Speakers:

Hyungryul Baik, KAIST, Korea

Sang-hyun Kim, Seoul National University, Korea

Thomas Koberda, Virginia University, USA

Seonhee Lim, Seoul National University, Korea

Kathryn Mann, Brown University, USA

Abstracts:

Hyungryul Baik, Mapping class group and the curve complex

We discuss the notion of stable translation length on the curve complex by elements of the mapping class group, and how we can use it to study various questions about mapping class groups and fibered 3-manifolds.

Sang-hyun Kim, RAAGs in Diff^r(S^1)

Let Diff^(S^1) denote the group of orientation-preserving C^r diffeomorphisms on the circle. In a joint work with Baik and Koberda, we proved that the group L = <a,b,c,d | [a,b]=[b,c]=[c,d] > does not embed into Diff^2(S^1), and deduced that most mapping class groups do not virtually admit a faithful C^2 action on the circle. This result will be summarized and generalized to classify all RAAGs in Diff^2(S^1), answering a question of Kharlamov. (Based on joint works Baik-Kim-Koberda and Kim-Koberda) .

Thomas Koberda, Commensurators of thin subgroups of PSL_2(Z)

A celebrated result of Margulis says that among irreducible lattices in higher rank semi-simple Lie groups, arithmetic lattices are characterized as those having dense commensurators. If the subgroup of the Lie group is Zariski dense and discrete but is no longer assumed to have finite covolume (that is, to be thin), then no such definitive dichotomy exists. A heuristic due to Y. Shalom says that thin subgroups should be thought of as non-arithmetic. In this talk I will discuss a theorem confirming Shalom's heuristic for certain naturally defined thin subgroups of PSL_2(Z). This is joint work with M. Mj and A. Reid.

Seonhee Lim, Continued fraction and extreme value distribution in trees

We show the extreme value distribution for the geodesic flow on the modular ray, the non-archimedean analog of the modular surface. One can use a Markov property to show that the measure of pointed geodesics which remains of distance at most u_n from the base point from time zero to n can be computed for certain function u_n. This is a joint work with Sanghoon Kwon.

Kathryn Mann, Rigidity of mapping class group actions on the circle

In this talk, I will describe a family of groups with the remarkable property that they have essentially only one nontrivial action on the circle by homeomorphisms. The examples are the mapping class groups of a surface S with a marked point. Such groups can be identified with the group of automorphisms of the fundamental group of S. In new joint work with M. Wolff, we show that any nontrivial action of such a group on the circle is semi-conjugate to its natural action on the Gromov boundary of \pi_1(S), answering a question of Farb. The talk will describe the proof of this result and some consequences related to regularity of group actions.

Program:

9:30 -  10:20 | Thomas Koberda, Commensurators of thin groups in PSL_2(Z)

10:30 - 11:20 | Seonhee Lim, Continued fraction and extreme value distribution in       trees

11:30 - 13:30 |  - lunch in the faculty restaurant -

14:00 - 14:50 | Kathryn Mann, Rigidity of mapping class group actions on the circle

15:00 - 15:50 | Hyungryul Baik, Mapping class group and the curve complex

15:50 - 16:20 | Coffee break

16:20 - 17:10 | Sang-hyun Kim, RAAGs in Diff^r(S^1) 

17:30 -       |  - dinner at Novotel-

Organizers: Nhan-Phu Chung, Sungkyunkwan university, Korea

Sponsored by: