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 \({n^{ - 3/5}} \ll p \ll {n^{ - 1/2 - \epsilon }}\) the fundamental group π 1(Y) has elements of order two and is of infinite cohomological dimension. We also prove that for \(p \ll {n^{ - 1/2 - \epsilon }}\) 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 \(p \ll {n^{ - 1/2 - \epsilon }}\)) 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).
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Costa, A.E., Farber, M. Geometry and topology of random 2-complexes. Isr. J. Math. 209, 883–927 (2015). https://doi.org/10.1007/s11856-015-1240-2
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DOI: https://doi.org/10.1007/s11856-015-1240-2