On the almost-palindromic width of free groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929456155721728 |
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| author | Staiger, Manuel |
| author_facet | Staiger, Manuel |
| contents | We answer a question of Bardakov (Kourovka Notebook, Problem 19.8) which asks for the existence of a pair of natural numbers $(c, m)$ with the property that every element in the free group on the two-element set $\{a, b\}$ can be represented as a concatenation of $c$, or fewer, $m$-almost-palindromes in letters $a^{\pm 1}, b^{\pm 1}$. Here, an $m$-almost-palindrome is a word which can be obtained from a palindrome by changing at most $m$ letters. We show that no such pair $(c, m)$ exists. In fact, we show that the analogous result holds for all non-abelian free groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15752 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the almost-palindromic width of free groups Staiger, Manuel Group Theory Combinatorics We answer a question of Bardakov (Kourovka Notebook, Problem 19.8) which asks for the existence of a pair of natural numbers $(c, m)$ with the property that every element in the free group on the two-element set $\{a, b\}$ can be represented as a concatenation of $c$, or fewer, $m$-almost-palindromes in letters $a^{\pm 1}, b^{\pm 1}$. Here, an $m$-almost-palindrome is a word which can be obtained from a palindrome by changing at most $m$ letters. We show that no such pair $(c, m)$ exists. In fact, we show that the analogous result holds for all non-abelian free groups. |
| title | On the almost-palindromic width of free groups |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2306.15752 |