Recurrence Relations for Cosets in Free Groups
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912716791218176 |
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| author | Reilly, Michael Shields, Cory |
| author_facet | Reilly, Michael Shields, Cory |
| contents | Let $F_2$ be the free group on two generators and let $H$ be a subgroup of $F_2$. We investigate a method for calculating the number of elements in a coset of $H$ that have a given length when written in reduced form. More specifically, taking $S_n\subseteq F_2$ to be the set of elements of length $n$, we show that for any coset $yH$ there always exists a recurrence relation of the form \[
|yH\cap S_n| = \sum_{i=1}^{n-1}\sum_{xH\in F_2/H}a_{i,xH}\cdot |xH\cap S_{n-i}| \] for some constants $(a_{i,xH})_{i\in \mathbb{N}, xH\in F_2/H}$, and we give an algorithm that calculates these constants. Further, we show that when $H$ has finite index and contains an element of odd length, only finitely many of the constants $a_{i,xH}$ are nonzero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14703 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recurrence Relations for Cosets in Free Groups Reilly, Michael Shields, Cory Group Theory 20E05 (Primary) 20E07 (Secondary) Let $F_2$ be the free group on two generators and let $H$ be a subgroup of $F_2$. We investigate a method for calculating the number of elements in a coset of $H$ that have a given length when written in reduced form. More specifically, taking $S_n\subseteq F_2$ to be the set of elements of length $n$, we show that for any coset $yH$ there always exists a recurrence relation of the form \[ |yH\cap S_n| = \sum_{i=1}^{n-1}\sum_{xH\in F_2/H}a_{i,xH}\cdot |xH\cap S_{n-i}| \] for some constants $(a_{i,xH})_{i\in \mathbb{N}, xH\in F_2/H}$, and we give an algorithm that calculates these constants. Further, we show that when $H$ has finite index and contains an element of odd length, only finitely many of the constants $a_{i,xH}$ are nonzero. |
| title | Recurrence Relations for Cosets in Free Groups |
| topic | Group Theory 20E05 (Primary) 20E07 (Secondary) |
| url | https://arxiv.org/abs/2511.14703 |