Phase transition for the existence of van Kampen 2-complexes in random groups
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912159313690624 |
|---|---|
| author | Tsai, Tsung-Hsuan |
| author_facet | Tsai, Tsung-Hsuan |
| contents | Gromov showed that (1993) with high probability, every bounded and reduced van Kampen diagram $D$ of a random group at density $d$ satisfies the isoperimetric inequality $|\partial D|\geq (1-2d-s)|D|\ell$. In this article, we adapt Gruber-Mackay's prove for random triangular groups, showing a non-reduced 2-complex version of this inequality.
Moreover, for any 2-complex $Y$ of a given geometric form, we exhibit a phase transition: we give explicitly a critical density $d_c$ depending only on $Y$ such that, in a random group at density $d$, if $d<d_c$ then there is no reduced van Kampen 2-complex of the form $Y$; while if $d>d_c$ then there exists reduced van Kampen 2-complexes of the form $Y$.
As an application, we show a phase transition for the $C(p)$ small-cancellation condition: for a random group at density $d$, if $d<1/(p+1)$ then it satisfies $C(p)$; while if $d>1/(p+1)$ then it does not satisfy $C(p)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_08234 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Phase transition for the existence of van Kampen 2-complexes in random groups Tsai, Tsung-Hsuan Group Theory 20F05, 20F06, 20P05 Gromov showed that (1993) with high probability, every bounded and reduced van Kampen diagram $D$ of a random group at density $d$ satisfies the isoperimetric inequality $|\partial D|\geq (1-2d-s)|D|\ell$. In this article, we adapt Gruber-Mackay's prove for random triangular groups, showing a non-reduced 2-complex version of this inequality. Moreover, for any 2-complex $Y$ of a given geometric form, we exhibit a phase transition: we give explicitly a critical density $d_c$ depending only on $Y$ such that, in a random group at density $d$, if $d<d_c$ then there is no reduced van Kampen 2-complex of the form $Y$; while if $d>d_c$ then there exists reduced van Kampen 2-complexes of the form $Y$. As an application, we show a phase transition for the $C(p)$ small-cancellation condition: for a random group at density $d$, if $d<1/(p+1)$ then it satisfies $C(p)$; while if $d>1/(p+1)$ then it does not satisfy $C(p)$. |
| title | Phase transition for the existence of van Kampen 2-complexes in random groups |
| topic | Group Theory 20F05, 20F06, 20P05 |
| url | https://arxiv.org/abs/2210.08234 |