# Victor Kostyuk

### First Position

Knowledge engineering at Reasoning Mind### Dissertation

*Outer Space for Two-Dimensional RAAGs and Fixed Point Sets of Finite Subgroups*

### Advisor

### Research Area

### Abstract

In [CCV07], Charney, Crisp, and Vogtmann construct an outer space for a 2-dimensional right-angled Artin group *A*_{Γ}. It is a contractible space on which a finite index subgroup Out^{0}(*A*_{Γ}) of Out(*A*_{Γ}) acts properly. We construct a different outer space S(*A*_{Γ}) for *A*_{Γ} and show that non-empty fixed point sets of finite subgroups of Out^{0}(*A*_{Γ}) are contractible in this space. While Culler’s realization theorem ([Cul84]) implies that fixed point sets of finite subgroups of Out(*Fn*) are always non-empty in the Culler-Vogtmann outer space, there is no direct counterpart to this result in the case of right-angled Artin groups and S(*A*_{Γ}). We present some methods for constructing elements in fixed point sets of finite subgroups and examine cases where such methods are applicable.