# Abstracts

# On the finiteness property of hyperbolic simplicial actions: The right-angled Artin groups and their extension graphs (2021)

We call a group G a *hyperbolic group* If a group admits a Gromov-hyperbolic Cayley graph. Such a group have interesting geometric or combinatorial property. As an example, Gromov showed that every element of a hyperbolic group has rational *asymptotic translation length* with respect to the action on the Cayley graph. Furthermore, the length-spectrum is always discrete, that is, there exists a common denominator shared by all asymptotic translation lengths. In 2010, Bowditch showed that the mapping class group action on the curve complex also has discrete and rational length-spectrum. On the other hand, right-angled Artin groups are not hyperbolic but have properties parallel to mapping class groups. In this article, we showed that the right-angled Artin group of a large-girth graph with the action on the extension graph has a rational and discrete length-spectrum. This is joint work with H. Baik and H. Shin. (arXiv:2103.13983)

# Powers of Dehn twists generating right-angled Artin groups (2021)

*Algebr. Geom. Topol.* 21 (2021), no. 3, 1511--1533

Dehn twists are the most elementary self-homeomorphism of a surface. Koberda showed that highly large powers of Dehn twists (plus subsurface pseudo-Anosov's) generate a right-angled Artin group (RAAG) as a subgroup of the mapping class group. In this article, we answered the question how large exponents of powers are sufficient to generate a RAAG. (arXiv:1989.03394)