Fundamental groups of clique complexes of random graphs

Armindo Costa, Michael Farber, Danijela Horak

Research output: Contribution to journalArticlepeer-review


We study fundamental groups of clique complexes associated to random Erdős–Rényi graphs (Formula presented.). We establish thresholds for a number of properties of fundamental groups of these complexes (Formula presented.). In particular, if (Formula presented.), then we show that (Formula presented.) asymptotically almost surely (a.a.s.), where (Formula presented.) and (Formula presented.) denote the geometric dimension and cohomological dimension correspondingly. It is known that the fundamental group (Formula presented.) is trivial for (Formula presented.). We prove that for (Formula presented.) the fundamental group (Formula presented.) has 2-torsion but has no (Formula presented.) -torsion for any given prime (Formula presented.). We also prove that aspherical subcomplexes of the random clique complex (Formula presented.) satisfy the Whitehead conjecture, that is, all their subcomplexes are also aspherical, a.a.s.

Original languageEnglish
Pages (from-to)1-32
Number of pages32
JournalTransactions of the London Mathematical Society
Issue number1
StatePublished - 2015
Externally publishedYes


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