In this talk we discuss completely syndetic (CS) sets in discrete groups - subsets that for every n admit finitely many left translates that jointly cover every n-tuple of group elements. While for finitely-generated groups, the non-virtually nilpotent ones admit a partition into two CS sets, we show that virtually abelian groups do not. We also characterize CS subsets of the integers, and as a result characterize subsets of integers whose closure in the Stone-Cech compacitifcation of Z contains the smallest two sided ideal. Finally, we show that CS sets can have an arbitrarily small density.

This talk is based upon arXiv:2506.18784, a joint work with Guy Salomon and Ariel Yadin.

A hyperbolic surface is called geometrically finite if it can be decomposed into a compact core together with finitely many cusps and funnels. The resolvent of the Laplacian on such spaces can be meromorphically continued to the entire complex plane. The poles of this meromorphic family of operators are called resonances and they contain a lot of information about the surface (for example about the long term behavior of solutions to the wave equation). We study such questions in the setting of "geometrically finite" regular graphs, which in particular requires us to make sense of what funnels and cusps mean in this setting. This is based on joint work with Christian Arends, Bartosz Trojan, and Tobias Weich.

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