Minicourse on right-angled Artin groups and their subgroups
Time: 2015-07-14
Published By:
Speaker(s): Thomas Koberda (Yale University)
Time: July 14 - July 16, 2015
Venue: Room 09 at Quan Zhai, BICMR
Speaker: Thomas Koberda (Yale University)
Time:
14:00-16:00 on 7.14 and 7.21.
9:00-11:00 on 7.16 and 7.23.
Venue: Room 9 at Quan Zhai, BICMR
Abstract:
The minicourse consists of four 2-hour lectures:
1. Introduction to right-angled Artin groups and mapping class groups.
This lecture will be a very basic introduction to right-angled Artin groups and mapping class groups. I will give definitions, foundational theorems, and lots of examples. Topics will include automorphisms of right-angled Artin groups, basics on braid groups, and the Nielsen-Thurston classification of mapping classes.
2. Right-angled Artin subgroups of mapping class groups.
In this lecture, I will discuss in detail how one can understand the right-angled Artin subgroups of mapping class groups, and some of the consequences. I will discuss the intimate connections between the algebra of right-angled Artin groups, dynamics in mapping class groups, and hyperbolic geometry which together allow one to establish the results I will discuss. Part of this lecture will discuss joint work with Sang-hyun Kim.
3. The extension graph, part 1: algebra.
In this lecture, I will define the extension graph and discuss some of its algebraic consequences, most importantly its usefulness for deciding whether there exists an injective homomorphism between two right-angled Artin groups. I will illustrate the theory further by drawing parallels between the extension graph for a right-angled Artin group and the curve graph for a surface. The content of this lecture will be joint work with Sang-hyun Kim.
4. The extension graph, part 2: geometry and dynamics.
In this lecture, I will develop the analogy between the extension graph and the curve graph, concentrating on geometry and dynamics. I will discuss the large-scale geometry of the extension graph, and the dynamics of the canonical right-angled Artin group action on the extension graph. The lecture will culminate in the complete description of finitely generated purely loxodromic subgroups of the right-angled Artin group, which will establish a result analogous to several conjectures on convex cocompactness in mapping class groups. Part of this lecture will represent joint work with Sang-hyun Kim, and part will represent joint work with Johanna Mangahas and Samuel Taylor.
Time:
14:00-16:00 on 7.14 and 7.21.
9:00-11:00 on 7.16 and 7.23.
Venue: Room 9 at Quan Zhai, BICMR
Abstract:
The minicourse consists of four 2-hour lectures:
1. Introduction to right-angled Artin groups and mapping class groups.
This lecture will be a very basic introduction to right-angled Artin groups and mapping class groups. I will give definitions, foundational theorems, and lots of examples. Topics will include automorphisms of right-angled Artin groups, basics on braid groups, and the Nielsen-Thurston classification of mapping classes.
2. Right-angled Artin subgroups of mapping class groups.
In this lecture, I will discuss in detail how one can understand the right-angled Artin subgroups of mapping class groups, and some of the consequences. I will discuss the intimate connections between the algebra of right-angled Artin groups, dynamics in mapping class groups, and hyperbolic geometry which together allow one to establish the results I will discuss. Part of this lecture will discuss joint work with Sang-hyun Kim.
3. The extension graph, part 1: algebra.
In this lecture, I will define the extension graph and discuss some of its algebraic consequences, most importantly its usefulness for deciding whether there exists an injective homomorphism between two right-angled Artin groups. I will illustrate the theory further by drawing parallels between the extension graph for a right-angled Artin group and the curve graph for a surface. The content of this lecture will be joint work with Sang-hyun Kim.
4. The extension graph, part 2: geometry and dynamics.
In this lecture, I will develop the analogy between the extension graph and the curve graph, concentrating on geometry and dynamics. I will discuss the large-scale geometry of the extension graph, and the dynamics of the canonical right-angled Artin group action on the extension graph. The lecture will culminate in the complete description of finitely generated purely loxodromic subgroups of the right-angled Artin group, which will establish a result analogous to several conjectures on convex cocompactness in mapping class groups. Part of this lecture will represent joint work with Sang-hyun Kim, and part will represent joint work with Johanna Mangahas and Samuel Taylor.
