TY - JOUR

T1 - Fundamental groups of clique complexes of random graphs

AU - Costa, Armindo

AU - Farber, Michael

AU - Horak, Danijela

N1 - Publisher Copyright:
© 2015 Author(s)

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84973929452&partnerID=8YFLogxK

U2 - 10.1112/tlms/tlv001

DO - 10.1112/tlms/tlv001

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AN - SCOPUS:84973929452

SN - 2052-4986

VL - 2

SP - 1

EP - 32

JO - Transactions of the London Mathematical Society

JF - Transactions of the London Mathematical Society

IS - 1

ER -