The infinite random simplicial complex

TL;DR

This paper explores the properties of the Fraisse limit of finite simplicial complexes, revealing model-theoretic results, subgroup embeddings, and a surprisingly simple topological structure of its geometric realisation.

Contribution

It introduces the notion of a local class, extends Fraisse theory to infinite languages, and characterizes the topological nature of the geometric realisation of the limit.

Findings

Existence of a 0-1 law for local classes

Every direct limit of finite groups embeds into the automorphism group

The geometric realisation is homeomorphic to the infinite simplex

Abstract

We study the Fraisse limit of the class of all finite simplicial complexes. Whilst the natural model-theoretic setting for this class uses an infinite language, a range of results associated with Fraisse limits of structures for finite languages carry across to this important example. We introduce the notion of a local class, with the class of finite simplicial complexes as an archetypal example, and in this general context prove the existence of a 0-1 law and other basic model-theoretic results. Constraining to the case where all relations are symmetric, we show that every direct limit of finite groups, and every metrizable profinite group, appears as a subgroup of the automorphism group of the Fraisse limit. Finally, for the specific case of simplicial complexes, we show that the geometric realisation is topologically surprisingly simple: despite the combinatorial complexity of the Fraisse limit, its geometric realisation is homeomorphic to the infinite simplex.