Exercise 5, § 6

Let G be the subgroup of $\mathbf{G}\mathbf{L}(2,\mathbf{R})$ formed by the matrices of positive determinant and let $\mathbf{H}$ be the set of complex numbers having strictly positive imaginary part (“Poincaré half-plane”). Put $\Gamma =\mathbf{S}\mathbf{L}(2,\mathbf{Z})$.

a) Prove that one makes G operate on $\mathbf{H}$ by putting

b) Let M be the set of complex numbers $z=x+iy\in \mathbf{H}$ such that $0\leqslant x\leqslant 1$ and $(x-1)^2+y^2\geqslant 1$. Prove that the family $(g\cdot M)_{g\in\Gamma}$ is a locally finite covering of $\mathbf{H}$ by closed subsets.

c) Put A = $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ and B = $\begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}$ . Verify that $A^2$ = $B^3=-I$. Prove that $\langle A,B,C ; A^2= B^3= C,C^2= 1\rangle$ is a presentation of the group Γ. (Apply prop. 10 of I, p. 136.)

d) Prove that the quotient group $\Gamma /\{\pm I\}$ has the presentation $\langle A,B ; A^2= B^3= 1\rangle$.