Snowflake groups and conjugator length functions with non-integer exponents
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914203338539008 |
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| author | Bridson, Martin R. Riley, Timothy R. |
| author_facet | Bridson, Martin R. Riley, Timothy R. |
| contents | We exhibit novel geometric phenomena in the study of conjugacy problems for discrete groups. We prove that the snowflake groups $B_{pq}$, indexed by pairs of positive integers $p>q$, have conjugator length functions $\text{CL}(n)\simeq n$ and annular Dehn functions $\text{Ann}(n) \simeq n^{2α}$, where $α= \log_2(2p/q)$. Then, building on $B_{pq}$, we construct groups $\tilde{B}_{pq}^+$, for which $\text{CL}(n)\simeq n^{α+1}$. Thus the conjugator length spectrum and the spectrum of exponents of annular Dehn functions are both dense in the range $[2,\infty)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14038 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Snowflake groups and conjugator length functions with non-integer exponents Bridson, Martin R. Riley, Timothy R. Group Theory 20F65 (Primary), 20F10 (Secondary) We exhibit novel geometric phenomena in the study of conjugacy problems for discrete groups. We prove that the snowflake groups $B_{pq}$, indexed by pairs of positive integers $p>q$, have conjugator length functions $\text{CL}(n)\simeq n$ and annular Dehn functions $\text{Ann}(n) \simeq n^{2α}$, where $α= \log_2(2p/q)$. Then, building on $B_{pq}$, we construct groups $\tilde{B}_{pq}^+$, for which $\text{CL}(n)\simeq n^{α+1}$. Thus the conjugator length spectrum and the spectrum of exponents of annular Dehn functions are both dense in the range $[2,\infty)$. |
| title | Snowflake groups and conjugator length functions with non-integer exponents |
| topic | Group Theory 20F65 (Primary), 20F10 (Secondary) |
| url | https://arxiv.org/abs/2512.14038 |