# 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 $$p,q>2$$