Orbit-blocking words and the average-case complexity of Whitehead's problem in the free group of rank 2
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913501165912064 |
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| author | Hyde, Lucy O'Connor, Siobhan Shpilrain, Vladimir |
| author_facet | Hyde, Lucy O'Connor, Siobhan Shpilrain, Vladimir |
| contents | Let F_2 denote the free group of rank 2. Our main technical result of independent interest is: for any element u of F_2, there is g in F_2 such that no cyclically reduced image of u under an automorphism of F_2 contains g as a subword. We then address computational complexity of the following version of the Whitehead automorphism problem: given a fixed u in F_2, decide, on an input v in F_2 of length n, whether or not v is an automorphic image of u. We show that there is an algorithm that solves this problem and has constant (i.e., independent of n) average-case complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12322 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orbit-blocking words and the average-case complexity of Whitehead's problem in the free group of rank 2 Hyde, Lucy O'Connor, Siobhan Shpilrain, Vladimir Group Theory Computational Complexity 20F10, 20F28 Let F_2 denote the free group of rank 2. Our main technical result of independent interest is: for any element u of F_2, there is g in F_2 such that no cyclically reduced image of u under an automorphism of F_2 contains g as a subword. We then address computational complexity of the following version of the Whitehead automorphism problem: given a fixed u in F_2, decide, on an input v in F_2 of length n, whether or not v is an automorphic image of u. We show that there is an algorithm that solves this problem and has constant (i.e., independent of n) average-case complexity. |
| title | Orbit-blocking words and the average-case complexity of Whitehead's problem in the free group of rank 2 |
| topic | Group Theory Computational Complexity 20F10, 20F28 |
| url | https://arxiv.org/abs/2406.12322 |