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Phase transition for
the existence of van Kampen $2$–complexes in random
groups
Tsung-Hsuan Tsai |
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Algebraic & Geometric Topology 24 (2024) 3897–3917
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Abstract |
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Gromov (1993) showed that every reduced van Kampen diagram of a random group at density satisfies the isoperimetric inequality . Adapting Gruber and Mackay’s (2021) method for random triangular groups, we obtain a nonreduced van Kampen –complex version of this inequality. The main result of this article is a phase transition: given a geometric form of –complexes, we find a critical density such that, in a random group at density , if , then there is no reduced van Kampen –complex of the form ; while if , then there exists reduced van Kampen –complexes of the form . As an application, we exhibit phase transitions for small-cancellation conditions in random groups, giving explicitly the critical densities for the conditions , , and . |
Keywords
geometric group theory, random group, van Kampen diagram,
isoperimetric inequality, small cancellation theory
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Mathematical Subject Classification
Primary: 20F06
Secondary: 20F05, 20P05
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References |
Publication
Received: 23 November 2022
Revised: 4 September 2023
Accepted: 23 October 2023
Published: 9 December 2024
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Authors |
| Tsung-Hsuan Tsai | |
| Institut de Recherche Mathématique
Avancée Université de Strasbourg Strasbourg France |
Institut Camille Jordan Université Claude Bernard Lyon 1 Villeurbanne France |
| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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