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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2009.01412 |
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| _version_ | 1866929244535259136 |
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| author | Chen, Haimiao |
| author_facet | Chen, Haimiao |
| contents | For any integers $m,n$ with $m\ne 0$ and $n>0$, let $G_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[z^m,x][z^n,y]\rangle$; for any integers $m,n>0$, let $H_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[x^m,z^n][y,z]\rangle$. By investigating cohomology jump loci of irreducible ${\rm GL}(2,\mathbb{C})$-character varieties, we show: if $m,m'\ne 0$, $n,n'>0$ and $G_{m',n'}\cong G_{m,n}$, then $m=m',n=n'$; if $m,m',n,n'>0$ and $H_{m',n'}\cong H_{m,n}$, then $m'=m, n'=n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_01412 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Solving the isomorphism problems for two families of parafree groups Chen, Haimiao Group Theory Algebraic Topology 14M35, 20J05 For any integers $m,n$ with $m\ne 0$ and $n>0$, let $G_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[z^m,x][z^n,y]\rangle$; for any integers $m,n>0$, let $H_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[x^m,z^n][y,z]\rangle$. By investigating cohomology jump loci of irreducible ${\rm GL}(2,\mathbb{C})$-character varieties, we show: if $m,m'\ne 0$, $n,n'>0$ and $G_{m',n'}\cong G_{m,n}$, then $m=m',n=n'$; if $m,m',n,n'>0$ and $H_{m',n'}\cong H_{m,n}$, then $m'=m, n'=n$. |
| title | Solving the isomorphism problems for two families of parafree groups |
| topic | Group Theory Algebraic Topology 14M35, 20J05 |
| url | https://arxiv.org/abs/2009.01412 |