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Solving the isomorphism problems for two families of parafree groups

Haimiao Chen

Abstract

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 $m'=m, n'=n$.

Solving the isomorphism problems for two families of parafree groups

Abstract

For any integers with and , let denote the group presented by ; for any integers , let denote the group presented by . By investigating cohomology jump loci of irreducible -character varieties, we show: if , and , then ; if and , then .

Paper Structure

This paper contains 13 sections, 8 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. The $G$ family
  3. $\mathbf{z}=\mathbf{d}(\lambda,\zeta)$ with $\zeta^n=1\ne\lambda^{n}$
  4. $\kappa_2=0$
  5. $\kappa_2\ne 0$
  6. $\mathbf{z}=\zeta\mathbf{p}$ with $\zeta^n=1$
  7. The result
  8. The $H$ family
  9. $r=2$
  10. $r\ne 2$
  11. $\mathbf{v}_{22}\ne 1$
  12. $\mathbf{v}_{22}=1$
  13. The result

Key Result

Theorem 1.1

For any integers $m,m',n,n'$ with $m,m'\ne 0$ and $n,n'>0$, if $G_{m,n}\cong G_{m',n'}$, then $m=m'$ and $n=n'$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:solution-G']}
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 8 more