Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2509.08325 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909980736618496 |
|---|---|
| author | Khezeli, Ali |
| author_facet | Khezeli, Ali |
| contents | We prove that the product of any two infinite countable groups has fixed price one. This resolves a longstanding problem posed by Gaboriau. The proof uses the propagation method to construct a Poisson horoball process as a weak limit of a sequence of factors of iid. We then construct a low-cost graphing by showing that the resulting horoballs have a variant of the infinite touching property almost surely, if the metric and the other parameters of the construction are chosen carefully. A novelty is providing direct simple proofs that do not rely on sophisticated results like amenability and double-recurrence, which are used in related works. An essential tool for avoiding any growth conditions is the convergence in the sense of point processes of pointed closed subsets, which is a notion from stochastic geometry. Also, to manage the overlapping of the horoballs, a generalization of the induction lemma is presented for random multisets of a group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08325 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Products of Infinite Countable Groups Have Fixed Price One Khezeli, Ali Group Theory Probability We prove that the product of any two infinite countable groups has fixed price one. This resolves a longstanding problem posed by Gaboriau. The proof uses the propagation method to construct a Poisson horoball process as a weak limit of a sequence of factors of iid. We then construct a low-cost graphing by showing that the resulting horoballs have a variant of the infinite touching property almost surely, if the metric and the other parameters of the construction are chosen carefully. A novelty is providing direct simple proofs that do not rely on sophisticated results like amenability and double-recurrence, which are used in related works. An essential tool for avoiding any growth conditions is the convergence in the sense of point processes of pointed closed subsets, which is a notion from stochastic geometry. Also, to manage the overlapping of the horoballs, a generalization of the induction lemma is presented for random multisets of a group. |
| title | Products of Infinite Countable Groups Have Fixed Price One |
| topic | Group Theory Probability |
| url | https://arxiv.org/abs/2509.08325 |