Simplicity of $*$-algebras of non-Hausdorff $\mathbb{Z}_2$-multispinal groupoids

Abstract: We study simplicity of $C^*$-algebras arising from self-similar groups of $\mathbb{Z}_2$-multispinal type, a generalization of the Grigorchuk case whose simplicity was first proved by L. Clark, R. Exel, E. Pardo, C. Starling, and A. Sims in 2019, and we prove results generalizing theirs. Our first main result is a sufficient condition for simplicity of the Steinberg algebra satisfying conditions modeled on the behavior of the groupoid associated to the first Grigorchuk group. This closely resembles conditions found by B. Steinberg and N. Szak\'acs. As a key ingredient we identify an infinite family of $2-(2q-1,q-1,q/2-1)$-designs, where $q$ is a positive even integer. We then deduce the simplicity of the associated $C^*$-algebra, which is our second main result. Results of similar type were considered by B. Steinberg and N. Szak\'acs in 2021, and later by K. Yoshida, but their methods did not follow the original methods of the five authors.