TY - JOUR

T1 - Geometry and topology of random 2-complexes

AU - Costa, A. E.

AU - Farber, M.

N1 - Publisher Copyright:
© 2015, Hebrew University of Jerusalem.

PY - 2015/9/1

Y1 - 2015/9/1

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=84945892131&partnerID=8YFLogxK

U2 - 10.1007/s11856-015-1240-2

DO - 10.1007/s11856-015-1240-2

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84945892131

SN - 0021-2172

VL - 209

SP - 883

EP - 927

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 2

ER -