Equivariant Benjamini-Schramm Convergence of Simplicial Complexes and $\ell^2$-Multiplicities

Steffen Kionke; Michael Schrödl-Baumann

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arXiv · 1905.05658

Equivariant Benjamini-Schramm Convergence of Simplicial Complexes and $\ell^2$-Multiplicities

Steffen Kionke·Michael Schrödl-Baumann

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Abstract

We define a variant of Benjamini-Schramm convergence for finite simplicial complexes with the action of a fixed finite group G which leads to the notion of random rooted simplicial G-complexes. For every random rooted simplicial G-complex we define a corresponding $\ell^2$-homology and the $\ell^2$-multiplicity of an irreducible representation of G in the homology. The $\ell^2$-multiplicities generalize the $\ell^2$-Betti numbers and we show that they are continuous on the space of sofic random rooted simplicial G-complexes. In addition, we study induction of random rooted complexes and discuss the effect on $\ell^2$-multiplicities.

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BibTeX RIS

Steffen Kionke, Michael Schrödl-Baumann. 2019-05-14. Equivariant Benjamini-Schramm Convergence of Simplicial Complexes and $\ell^2$-Multiplicities. https://arxiv.org/abs/1905.05658

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