Spherical growth of reciprocal classes in the Hecke Groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910968207900672 |
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| author | Das, Debattam Gongopadhyay, Krishnendu |
| author_facet | Das, Debattam Gongopadhyay, Krishnendu |
| contents | Let $Γ_p$ denote the Hecke group where $p=2r$, $r>0$. Let $\mathcal{N}_l$ denote the set of conjugacy classes of reciprocal elements of word length $l$ in $Γ_p$. We prove that for $l \to \infty$,
$$|\mathcal{N}_l| = \mathcal{O}\left(\left\lfloor \tfrac{l+1}{2} \right\rfloor^{s-1} ρ^{\left\lfloor \tfrac{l+1}{2} \right\rfloor} \right),
$$
where $\mathcal O$ is the `big O', $ρ\in [\sqrt{2}, 2]$ is the unique positive real root of
$$
p(x) = x^{r+1} - 2\sum_{j=1}^{r-1} x^{r-j} - 1,
$$
and $s$ is the maximal multiplicity among the roots of $p(x)$.
Our method relies on the free product structure of the Hecke group $Γ_p$, a combinatorial counting function, and recurrence relations derived from cyclically reduced representatives. We also derive that the growth rate of the primitive reciprocal classes of word length $l$ is in agreement with that of $\mathcal{N}_l$. This work generalizes previous results for odd $p$ and provides an explicit asymptotic bound for all Hecke groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00739 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spherical growth of reciprocal classes in the Hecke Groups Das, Debattam Gongopadhyay, Krishnendu Group Theory Geometric Topology Primary 20H10, Secondary 11F06, 05E16, 37C35 Let $Γ_p$ denote the Hecke group where $p=2r$, $r>0$. Let $\mathcal{N}_l$ denote the set of conjugacy classes of reciprocal elements of word length $l$ in $Γ_p$. We prove that for $l \to \infty$, $$|\mathcal{N}_l| = \mathcal{O}\left(\left\lfloor \tfrac{l+1}{2} \right\rfloor^{s-1} ρ^{\left\lfloor \tfrac{l+1}{2} \right\rfloor} \right), $$ where $\mathcal O$ is the `big O', $ρ\in [\sqrt{2}, 2]$ is the unique positive real root of $$ p(x) = x^{r+1} - 2\sum_{j=1}^{r-1} x^{r-j} - 1, $$ and $s$ is the maximal multiplicity among the roots of $p(x)$. Our method relies on the free product structure of the Hecke group $Γ_p$, a combinatorial counting function, and recurrence relations derived from cyclically reduced representatives. We also derive that the growth rate of the primitive reciprocal classes of word length $l$ is in agreement with that of $\mathcal{N}_l$. This work generalizes previous results for odd $p$ and provides an explicit asymptotic bound for all Hecke groups. |
| title | Spherical growth of reciprocal classes in the Hecke Groups |
| topic | Group Theory Geometric Topology Primary 20H10, Secondary 11F06, 05E16, 37C35 |
| url | https://arxiv.org/abs/2411.00739 |