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 unimodular random rooted simplicial G-complexes. For every unimodular random rooted simplicial G-complex we define a corresponding ℓ2-homology and the ℓ2-multiplicity of an irreducible representation of G in the homology. The ℓ2-multiplicities generalize the ℓ2-Betti numbers and we show that they are continuous on the space of sofic random rooted simplicial G-complexes. In addition, we study the induction of random rooted complexes and discuss the effect on ℓ2-multiplicities.

中文翻译：

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单纯复形和 ℓ2-多重性的等变 Benjamini-Schramm 收敛

我们定义有限单纯复形的 Benjamini-Schramm 收敛变体，具有固定有限群的作用G这导致了单模随机根单纯形的概念G-复合体。对于每个单模随机有根单纯形G-complex 我们定义一个对应的ℓ2-同源性和ℓ2- 不可约表示的多重性G在同源性。这ℓ2- 多重性概括了ℓ2-Betti 数，我们证明它们在 sofic 随机根单纯形空间上是连续的G-复合体。此外，我们研究了随机根复合物的诱导并讨论了对ℓ2- 多重性。