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Complete Classification of Groups of Genus 1

The following is a summary of Proulx's classification of groups of genus 1. Because toroidal groups can only have two, three, or four generators, we're able to separate the 28 cases accordingly.

Two Generators

If \(\{x,y\}\) forms a generating set of a group \(G\) such that \(\gamma(G)\neq0\), then \(\gamma(G) = 1\) if and only if at least one of the following hold:
  1. \(G = \langle x,y\:|\: y^2 = x^3yxy = 1\rangle\)
  2. \(G = \langle x,y\:|\: y^2 = x^{-3}yxy = 1\rangle\)
  3. \(y^2 = (x^2y)^2 = 1\) and \(x\) and \(xy\) have even order, and the subgroup \(S\) generated by \(\{x^2, (xy)^2\}\) is normal and \(|G|/|S| = 4\)
  4. \(y^2 = x^2yx^{-2}y = 1\) and \(x\) is of even order, and the subgroup \(S\) generated by \(\{x^2,yxy\}\) is such that \(|G|/|S| = 2\)
  5. \(x^6 = y^2 = (xy)^3 = 1\)
  6. \(x^4 = y^2 = (xy)^4 = 1\)
  7. \(x^4 = y^2 = (xy)^2(x^{-1}y)^2 = 1\)
  8. \(x^3 = y^2 = (xyxyxy^{-1}x)^2 = 1\)
  9. \(x^3 = y^2 = (xy)^3(x^{-1}y)^3\)
  10. \(x^4 = y^2 = (xyx^{-1}y)^2 = 1\) and the subgroup \(S\) generated by \(\{x,yxy\}\) is such that \(|G|/|S|=2\)
  11. \(x^3 = y^2 = (xyx^{-1}y)^3 = 1\)
  12. \(G = \mathbb{Z}_m \times \mathbb{Z}_n,\,\,\gcd(m,n) \geq 3\)
  13. \(x^{2a} = y^b = xyx^{-1}y = 1\) for integers \(a>1\), \(b>2\), and the subgroup \(S\) generated by \(\{x^2, y\}\) is such that \(|G|/|S| = 2\)
  14. \(x^{2a} = y^{2b}=x^2y^2=1\) for integers \(a, b > 1\) and the subgroup \(S\) generated by \(\{x^{-1}y, x^2\}\) is such that \(|G|/|S| = 2\)
  15. \(x^{2a} = y^{2b} = (xy)^2 = (x^{-1}y)^2 = 1\) for integers \(a,b>1\) and the subgroup \(S\) generated by \(\{x^2,y^2\}\) is such that \(|G|/|S| = 4\)
  16. \(x^3 = y^3 = (xy)^3 = 1\)
  17. \(G = \langle x,y\:|\: x^3 = y^3 = xyxy^{-1}x^{-1}y^{-1} = 1\rangle\)

Three Generators

If \(\{x,y,z\}\) forms a generating set of a group \(G\) such that \(\gamma(G)\neq0\), then \(\gamma(G) = 1\) if and only if at least one of the following hold:
  1. \(x^2=y^2=z^2=(xy)^3=(xz)^3=(yz)^3 = 1\) and all other reduced relations are have even length
  2. \(x^2 = y^2=z^2 = (xyz)^2 = 1\)
  3. \(x^2=y^2=z^2=xyxzyz=(xy)^{2a}=(zy)^{2a}=1\) for integer \(a>0\) and the subgroup generated by \(\{xz,zyzy\}\) is normal and is such that \(|G|/|S| = 4\)
  4. \(x^2=y^2=z^2=(zxzy)^2 = (xy)^2 = 1\)
  5. \(x^2=y^2=z^2=(xz)^2(yz)^2 = (xy)^2 = 1\)
  6. \(x^2=y^2=z^2= (xy)^2 = (xz)^4 = (yz)^4 = 1\) and the subgroup \(S\) generated by \(\{xy, xz, yz\}\) is such that \(|G|/|S| = 2\)
  7. \(x^2=y^2=z^2= (xy)^2 = (xz)^6 = (yz)^3 = 1\) and the subgroup \(S\) generated by \(\{xy, xz, yz\}\) is such that \(|G|/|S| = 2\)
  8. \(x^a = y^2 = z^2 = (xy)^2 = (xz)^2 = (yz)^b = 1\) for relatively prime integers \(a, b\), \(a>2\) and the subgroup \(S\) generated by \{x, yz\} is such that \(|G|/|S| = 2\)
  9. \(x^{2a}=y^2=z^2= (xy)^2 = xzx^{-1}z = (yz)^{2b} = 1\) for \(a,b>1\) and the subgroup \(S\) generated by \(\{x, (yz)^2\}\) is normal and such that \(|G|/|S| = 2\)
  10. \(x^a = y^2 = z^2 = (xy)^2 = xyx^{-1}y = xzx^{-1}z =(yz)^b= 1\) for relatively prime integers \(a, b\), \(a>2\) and the subgroup \(S\) generated by \{x, yz\} is such that \(|G|/|S| = 2\)

Four Generators

If \(\{x,y,z,w\}\) forms a generating set of a group \(G\) such that \(\gamma(G)\neq0\), then \(\gamma(G) = 1\) if and only if
  1. \(x^2=y^2=z^2=w^2=(xy)^2=(yz)^2=(zw)^2=(wx)^2=(xz)^a=(yw)^b=1\) where \(a=2\) and \(q\) is even, or \(p,q>2\)