Geometry and topology of random 2-complexes

A. E. Costa*, M. Farber

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We study random 2-dimensional complexes in the Linial-Meshulam model and prove that the fundamental group of a random 2-complex Y has cohomological dimension ≤ 2 if the probability parameter satisfies p ≪ n−3/5. Besides, for (Formula Presented.) the fundamental group π1(Y) has elements of order two and is of infinite cohomological dimension. We also prove that for (Formula Presented.) the fundamental group of a random 2-complex has no m-torsion, for any given odd prime m ≥ 3. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be aspherical; this implies that (for (Formula Presented.)) any aspherical subcomplex of a random 2-complex satisfies the Whitehead conjecture. We use inequalities for Cheeger constants and systoles of simplicial surfaces to analyse spheres and projective planes lying in random 2-complexes. Our proofs exploit the uniform hyperbolicity property of random 2-complexes (Theorem 3.4).

Original languageEnglish
Pages (from-to)883-927
Number of pages45
JournalIsrael Journal of Mathematics
Volume209
Issue number2
DOIs
StatePublished - 1 Sep 2015
Externally publishedYes

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