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Gromov (1993) showed that every reduced van Kampen diagram D of a random group at density d satisfies the isoperimetric inequality |∂D|≥ (1 − 2d − 𝜀)|D|ℓ. Adapting Gruber and Mackay’s (2021) method for random triangular groups, we obtain a nonreduced van Kampen 2–complex version of this inequality.
The main result of this article is a phase transition: given a geometric form Y of 2–complexes, we find a critical density dc(Y ) such that, in a random group at density d, if d < dc, then there is no reduced van Kampen 2–complex of the form Y ; while if d > dc, then there exists reduced van Kampen 2–complexes of the form Y .
As an application, we exhibit phase transitions for small-cancellation conditions in random groups, giving explicitly the critical densities for the conditions C′(λ), C(p), B(p) and T(q).
Tsai, Tsung-Hsuan 1
@article{10_2140_agt_2024_24_3897,
author = {Tsai, Tsung-Hsuan},
title = {Phase transition for the existence of van {Kampen} 2{\textendash}complexes in random groups},
journal = {Algebraic and Geometric Topology},
pages = {3897--3917},
publisher = {mathdoc},
volume = {24},
number = {7},
year = {2024},
doi = {10.2140/agt.2024.24.3897},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.3897/}
}
TY - JOUR AU - Tsai, Tsung-Hsuan TI - Phase transition for the existence of van Kampen 2–complexes in random groups JO - Algebraic and Geometric Topology PY - 2024 SP - 3897 EP - 3917 VL - 24 IS - 7 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.3897/ DO - 10.2140/agt.2024.24.3897 ID - 10_2140_agt_2024_24_3897 ER -
%0 Journal Article %A Tsai, Tsung-Hsuan %T Phase transition for the existence of van Kampen 2–complexes in random groups %J Algebraic and Geometric Topology %D 2024 %P 3897-3917 %V 24 %N 7 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.3897/ %R 10.2140/agt.2024.24.3897 %F 10_2140_agt_2024_24_3897
Tsai, Tsung-Hsuan. Phase transition for the existence of van Kampen 2–complexes in random groups. Algebraic and Geometric Topology, Tome 24 (2024) no. 7, pp. 3897-3917. doi: 10.2140/agt.2024.24.3897
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