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

{x, y}

forms a generating set of a group

G

such that

γ (G) \neq 0

, then

γ (G) = 1

if and only if at least one of the following hold:

$G = ⟨ x, y | y^{2} = x^{3} y x y = 1 ⟩$
$G = ⟨ x, y | y^{2} = x^{- 3} y x y = 1 ⟩$
$y^{2} = (x^{2} y)^{2} = 1$ and $x$ and $x y$ have even order, and the subgroup $S$ generated by ${x^{2}, (x y)^{2}}$ is normal and $| G | / | S | = 4$
$y^{2} = x^{2} y x^{- 2} y = 1$ and $x$ is of even order, and the subgroup $S$ generated by ${x^{2}, y x y}$ is such that $| G | / | S | = 2$
$x^{6} = y^{2} = (x y)^{3} = 1$
$x^{4} = y^{2} = (x y)^{4} = 1$
$x^{4} = y^{2} = (x y)^{2} (x^{- 1} y)^{2} = 1$
$x^{3} = y^{2} = (x y x y x y^{- 1} x)^{2} = 1$
$x^{3} = y^{2} = (x y)^{3} (x^{- 1} y)^{3}$
$x^{4} = y^{2} = (x y x^{- 1} y)^{2} = 1$ and the subgroup $S$ generated by ${x, y x y}$ is such that $| G | / | S | = 2$
$x^{3} = y^{2} = (x y x^{- 1} y)^{3} = 1$
$G = Z_{m} \times Z_{n}, gcd (m, n) \geq 3$
$x^{2 a} = y^{b} = x y x^{- 1} y = 1$ for integers $a > 1$ , $b > 2$ , and the subgroup $S$ generated by ${x^{2}, y}$ is such that $| G | / | S | = 2$
$x^{2 a} = y^{2 b} = x^{2} y^{2} = 1$ for integers $a, b > 1$ and the subgroup $S$ generated by ${x^{- 1} y, x^{2}}$ is such that $| G | / | S | = 2$
$x^{2 a} = y^{2 b} = (x y)^{2} = (x^{- 1} y)^{2} = 1$ for integers $a, b > 1$ and the subgroup $S$ generated by ${x^{2}, y^{2}}$ is such that $| G | / | S | = 4$
$x^{3} = y^{3} = (x y)^{3} = 1$
$G = ⟨ x, y | x^{3} = y^{3} = x y x y^{- 1} x^{- 1} y^{- 1} = 1 ⟩$

Three Generators

{x, y, z}

forms a generating set of a group

G

such that

γ (G) \neq 0

, then

γ (G) = 1

if and only if at least one of the following hold:

$x^{2} = y^{2} = z^{2} = (x y)^{3} = (x z)^{3} = (y z)^{3} = 1$ and all other reduced relations are have even length
$x^{2} = y^{2} = z^{2} = (x y z)^{2} = 1$
$x^{2} = y^{2} = z^{2} = x y x z y z = (x y)^{2 a} = (z y)^{2 a} = 1$ for integer $a > 0$ and the subgroup generated by ${x z, z y z y}$ is normal and is such that $| G | / | S | = 4$
$x^{2} = y^{2} = z^{2} = (z x z y)^{2} = (x y)^{2} = 1$
$x^{2} = y^{2} = z^{2} = (x z)^{2} (y z)^{2} = (x y)^{2} = 1$
$x^{2} = y^{2} = z^{2} = (x y)^{2} = (x z)^{4} = (y z)^{4} = 1$ and the subgroup $S$ generated by ${x y, x z, y z}$ is such that $| G | / | S | = 2$
$x^{2} = y^{2} = z^{2} = (x y)^{2} = (x z)^{6} = (y z)^{3} = 1$ and the subgroup $S$ generated by ${x y, x z, y z}$ is such that $| G | / | S | = 2$
$x^{a} = y^{2} = z^{2} = (x y)^{2} = (x z)^{2} = (y z)^{b} = 1$ for relatively prime integers $a, b$ , $a > 2$ and the subgroup $S$ generated by \{x, yz\} is such that $| G | / | S | = 2$
$x^{2 a} = y^{2} = z^{2} = (x y)^{2} = x z x^{- 1} z = (y z)^{2 b} = 1$ for $a, b > 1$ and the subgroup $S$ generated by ${x, (y z)^{2}}$ is normal and such that $| G | / | S | = 2$
$x^{a} = y^{2} = z^{2} = (x y)^{2} = x y x^{- 1} y = x z x^{- 1} z = (y z)^{b} = 1$ for relatively prime integers $a, b$ , $a > 2$ and the subgroup $S$ generated by \{x, yz\} is such that $| G | / | S | = 2$

Four Generators

{x, y, z, w}

forms a generating set of a group

G

such that

γ (G) \neq 0

, then

γ (G) = 1

if and only if

$x^{2} = y^{2} = z^{2} = w^{2} = (x y)^{2} = (y z)^{2} = (z w)^{2} = (w x)^{2} = (x z)^{a} = (y w)^{b} = 1$ where $a = 2$ and $q$ is even, or $p, q > 2$