Random semigroups á la Gromov
Abstract
We introduce a model of random semigroups analogous to Gromov's density model for random groups. For a fixed number n≥ 2 of generators and a density parameter δ∈ (0,1), we impose a uniformly chosen set of ⌊ n2δl⌋ relations of the form u=v, where u,v are distinct words of length at most l, and study the resulting semigroup as l→∞. We prove that, with overwhelming probability, above density 1/2 the semigroup is trivial, while below density 1/2 it contains a free subsemigroup of rank two, has a Golod-Shafarevich semigroup algebra over every field, and has a homomorphic image which generates a non-elementary hyperbolic group. We also show that the small cancellation condition C(m) holds with overwhelming probability whenever δ<1/2m. Finally, for n≥ 3, at critical density 1/2 we obtain a subsequential limiting distribution on finite semigroups, with positive mass at the trivial semigroup and unbounded support in size. This contrasts with random groups, which have order at most two with overwhelming probability at critical density.
Related
- Strong law of large numbers for random semigroups in norm topology
- A new hierarchy for automaton semigroups
- Kobayashi's problem for power semigroups of inverse semigroups
- Steiner Minimal Trees in the Gromov--Hausdorff Space and in Gromov--Hausdorff Clouds
- Lower curvature bounds on metric surfaces via the Bishop--Gromov inequality
BibTeX
@misc{blachar2026random,
title = {{Random semigroups á la Gromov}},
author = {Guy Blachar and Be'eri Greenfeld and Margaret Shideler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39673},
eprint = {2610.39673},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39673},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Random semigroups á la Gromov
AU - Guy Blachar
AU - Be'eri Greenfeld
AU - Margaret Shideler
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39673
UR - https://arxiv.org/abs/2610.39673
AB - We introduce a model of random semigroups analogous to Gromov's density model for random groups. For a fixed number n≥ 2 of generators and a density parameter δ∈ (0,1), we impose a uniformly chosen set of ⌊ n^2δl⌋ relations of the form u=v, where u,v are distinct words of length at most l, and study the resulting semigroup as l→∞. We prove that, with overwhelming probability, above density 1/2 the semigroup is trivial, while below density 1/2 it contains a free subsemigroup of rank two, has a Golod-Shafarevich semigroup algebra over every field, and has a homomorphic image which generates a non-elementary hyperbolic group. We also show that the small cancellation condition C(m) holds with overwhelming probability whenever δ<1/2m. Finally, for n≥ 3, at critical density 1/2 we obtain a subsequential limiting distribution on finite semigroups, with positive mass at the trivial semigroup and unbounded support in size. This contrasts with random groups, which have order at most two with overwhelming probability at critical density.
ER -