Geometric properties of the golden ration Thompson's group
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913137882562560 |
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| author | Svetelik, Denys |
| author_facet | Svetelik, Denys |
| contents | We show that all three golden ratio Thompson's groups $F_τ$, $T_τ$ and $V_τ$ embed in the asynchronous rational group. We prove properties of the Cayley graph of the monoid $M = \langle L, R : LR^2 = RL^2 \rangle$, whose topological full group is $V_τ$. In particular, we compute a distance function for the Cayley graph of the monoid $M$. Additionally, we prove that this Cayley graph is hyperbolic in the sense of Gromov. Our analysis reveals that the horofunction boundary of this graph is homeomorphic to a space resembling a Cantor-like set, with additional isolated points situated between each pair of breakpoints. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_17814 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric properties of the golden ration Thompson's group Svetelik, Denys Group Theory Metric Geometry 20F65, 51F99 We show that all three golden ratio Thompson's groups $F_τ$, $T_τ$ and $V_τ$ embed in the asynchronous rational group. We prove properties of the Cayley graph of the monoid $M = \langle L, R : LR^2 = RL^2 \rangle$, whose topological full group is $V_τ$. In particular, we compute a distance function for the Cayley graph of the monoid $M$. Additionally, we prove that this Cayley graph is hyperbolic in the sense of Gromov. Our analysis reveals that the horofunction boundary of this graph is homeomorphic to a space resembling a Cantor-like set, with additional isolated points situated between each pair of breakpoints. |
| title | Geometric properties of the golden ration Thompson's group |
| topic | Group Theory Metric Geometry 20F65, 51F99 |
| url | https://arxiv.org/abs/2605.17814 |