Freeness of the random fundamental group

Newman, Andrew

doi:10.1142/s1793525319500468

Cited by 5 publications

(6 citation statements)

References 19 publications

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“…We further believe it is possible that for p above this threshold, is free with high probability. For the case of with , the sharp threshold is found by Newman in [27], improving on previous work of [7].…”

Section: Introductionsupporting

confidence: 74%

Topology of random -dimensional cubical complexes

Kahle

Paquette

Roldán

2021

Forum of Mathematics, Sigma

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We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

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Section: Introductionsupporting

confidence: 74%

Topology of random -dimensional cubical complexes

Kahle

Paquette

Roldán

2021

Forum of Mathematics, Sigma

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“…Note that the algebraic structure of the fundamental group is more intricate than H 1 (X), and thus several other properties have been studied in the literature. For example: its geometric and cohomological dimension [90], torsion [91], property T [33], freeness [92], and more.…”

Section: The Fundamental Groupmentioning

confidence: 99%

Random Simplicial Complexes: Models and Phenomena

Bobrowski,

Krioukov

2021

Preprint

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We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results have been recently established rigorously in mathematics, especially in the context of algebraic topology. In application to real-world systems, the reviewed models are typically used as null models, so that we take a statistical stance, emphasizing, where applicable, the entropic properties of the reviewed models. We also review a collection of phenomena and features observed in these models, and split the presented results into two classes: phase transitions and distributional limits. We conclude with an outline of interesting future research directions.

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“…• X ∼ X(n, p) to go from π 1 (X) free to π 1 (X) nonfree. The constants c d and γ d are described fully in [16], where the following table of the approximation of the first few values also appears: Regarding the fundamental group, [18] proves the following for Y ∼ Y 2 (n, p). These are still the best known bounds on a sharp threshold for π 1 (Y 2 (n, p)) to be free.…”

Section: Discussionmentioning

confidence: 98%

“…[18] If c < γ 2 and Y ∼ Y 2 (n, c/n) then with high probability π 1 (Y ) is free, while if c > c 2 and Y ∼ Y 2 (n, c/n) then with high probability π 1 (Y ) is not free.Naturally Conjecture 22 together with Theorem 16 would imply that π 1 (X(n, p)) is free if p < √ γ n ≈ 1.567 √ n and if not free if p > √ c 2…”

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confidence: 98%

One-sided sharp thresholds for homology of random flag complexes

Newman¹

2021

Preprint

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We prove that the random flag complex has a probability regime where the probability of nonvanishing homology is asymptotically bounded away from zero and away from one. Related to this main result we also establish new bounds on a sharp threshold for the fundamental group of a random flag complex to be a free group. In doing so we show that there is an intermediate probability regime in which the random flag complex has fundamental group which is neither free nor has Kazhdan's property (T).

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Freeness of the random fundamental group

Cited by 5 publications

References 19 publications

Topology of random -dimensional cubical complexes

Topology of random -dimensional cubical complexes

Random Simplicial Complexes: Models and Phenomena

One-sided sharp thresholds for homology of random flag complexes

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