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- FIELD
- Math: HCMC
- DATE
-
Oct 04 (Fri), 2024
- TIME
- 16:00 ~ 17:00
- PLACE
- 1423
- SPEAKER
- Jang, Wonyong
- HOST
- Kwak, Sanghoon
- INSTITUTE
- KAIST
- TITLE
- [HG_T] On the kernel of group actions on asymptotic cones
- ABSTRACT
- The concept of an asymptotic cone was first suggested by Gromov and he used it to establish Gromov's polynomial growth theorem. An asymptotic cone of a group reflects many properties of the group. For example, a group is virtually nilpotent if and only if all of its asymptotic cones are locally compact (equivalently, proper). Also, a finitely generated group is hyperbolic if and only if all of its asymptotic cones are real trees.
In this talk, we characterize the natural kernel of the action of a group G on its asymptotic cone. Our main theorem states that if G is acylindrically hyperbolic, then the kernel of G-action on an asymptotic cone of G is the same as K(G), the unique maximal finite normal subgroup. Then it turns out that the kernel also coincides with many algebraically defined subgroups. Moreover, this result does not depend on the choice of ultrafilter and sequence that we need to define asymptotic cones so it implies that the kernel is invariant under the choice of these. It is known that a group may have distinct (actually, non-homeomorphic) asymptotic cones, and indeed some acylindrically hyperbolic groups have various asymptotic cones.
As an application, we relate this kernel to other kernels of group actions on other spaces at "infinity", for instance, the limit set of convergence group action, Floyd boundary, and many boundaries of CAT(0) spaces with some conditions. In addition, we will introduce another action of G on an asymptotic cone, called Paulin's construction, and describe the kernel of Paulin's construction. If time permits, we will prove that the kernel can determine whether a given action is non-elementary, under specific conditions, and give the kernel of several (not acylindrically hyperbolic) groups. This work is joint with my advisor, Hyungryul Baik.
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