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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.13644 |
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| _version_ | 1866916257294450688 |
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| author | Howie, James |
| author_facet | Howie, James |
| contents | A conjecture of Rosenberger says that a group of the form $\langle x,y|x^p=y^q=W(x,y)^r=1\rangle$ (with $r>1$) is either virtually solvable or contains a non-abelian free subgroup. This note is an account of an attack on the conjecture in the case $(p,q,r)=(2,4,2)$. The results obtained are only partial, but nevertheless provide strong evidence in support of the conjecture in the case in question, in that the word $W$ in any counterexample is shown to satisfy some strong restrictions. The exponent-sums of $x$ and $y$ in $W$ must be even and odd respectively, while its free-product (or syllable) length must be at least 68. There is also a report of computer investigations which yield a stronger lower bound of 196 for the free-product length. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13644 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalised Triangle Groups of Type (2,4,2) Howie, James Group Theory 20F05 (Primary) 20E05, 20C99, 20-08 (Secondary) A conjecture of Rosenberger says that a group of the form $\langle x,y|x^p=y^q=W(x,y)^r=1\rangle$ (with $r>1$) is either virtually solvable or contains a non-abelian free subgroup. This note is an account of an attack on the conjecture in the case $(p,q,r)=(2,4,2)$. The results obtained are only partial, but nevertheless provide strong evidence in support of the conjecture in the case in question, in that the word $W$ in any counterexample is shown to satisfy some strong restrictions. The exponent-sums of $x$ and $y$ in $W$ must be even and odd respectively, while its free-product (or syllable) length must be at least 68. There is also a report of computer investigations which yield a stronger lower bound of 196 for the free-product length. |
| title | Generalised Triangle Groups of Type (2,4,2) |
| topic | Group Theory 20F05 (Primary) 20E05, 20C99, 20-08 (Secondary) |
| url | https://arxiv.org/abs/2405.13644 |