Spring 2024

I'll provide an exposition of an article by J. Walton and M. Whittaker under the same title. Their abstract: "We show that Kellendonk's tiling semigroup of an FLC (finite local complexity) substitution tiling is self-similar, in the sense of Bartholdi, Grigorchuk, and Nekrashevych. We extend the notion of limit space of a self-similar group to the setting of self-similar semigroups, and show that it is homeomorphic to the Anderson-Putnam complex for such substitution tilings, with natural self-map induced by the substitution. Thus, the inverse limit of the limit space, given by the limit solenoid of the self-similar semigroup, is homeomorphic to the translational hull of the tiling."

I'll provide an exposition of an article by J. Walton and M. Whittaker under the same title. Their abstract: "We show that Kellendonk's tiling semigroup of an FLC (finite local complexity) substitution tiling is self-similar, in the sense of Bartholdi, Grigorchuk, and Nekrashevych. We extend the notion of limit space of a self-similar group to the setting of self-similar semigroups, and show that it is homeomorphic to the Anderson-Putnam complex for such substitution tilings, with natural self-map induced by the substitution. Thus, the inverse limit of the limit space, given by the limit solenoid of the self-similar semigroup, is homeomorphic to the translational hull of the tiling.

Given D a natural number and P a subgroup of the group of automorphisms of the finite d-regular rooted tree with D levels, the group of finite type GP is the subgroup of the automorphism of the infinite d-regular rooted tree whose elements only have sections in the subgroup P.

In the article "On finite generation of self-similar groups of finite type", Bondarenko and Samoilovych claim that in the case (d,D) = (2,4) there are 32 topologically finitely generated finite type groups and 20 of them are pairwise non-isomorphic.

I will introduce the notions of minimal Cantor systems, étale groupoids, and the topological full groups arising from them. Then I will introduce some interesting subgroups of the topological full groups. The talk is primarily an explanation of definitions with (hopefully) lots of examples presented. 

We will introduce Thompson's group V as defined in terms of diagrams, and we will introduce the notion of chromatic and eventually-chromatic subgroups of Thompson's group $V_2$. We will introduce the notion of a discrete automatic process, and we will formulate and (time permitting) prove a correspondence between the eventually-chromatic subgroups of $V_2$ and the transparent discrete automatic processes on an alphabet of two characters.

I'll present a characterization of hyperbolic groups as being those which are strongly geodesically automatic.