Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 3F / Solution

Let $H_k$ be the open hyperbolic half-plane beyond ${\gamma }_k$, opposite the central region $U$. These three half-planes have disjoint closures in the disk, and hyperbolic reflection $J_k$ exchanges $H_k$ and its complementary half-plane. In particular $J_k(U^{\circ })\subset H_k$ and $J_i(H_j)\subset H_i$ when $i\ne j$. A reduced nonempty reflection word therefore sends an interior point of $U$ into the half-plane corresponding to its first letter. It cannot be the identity. This ping-pong lemma argument proves that the reflection group isThe usual reflection tiling makes $U$ a fundamental domain for $W$: successive reflection across an offending side reduces a point to $U$, and the disjoint-side configuration yields locally finite tiles. Equivalently this is the disjoint-geodesic case of the reflection fundamental-domain theorem; there are no vertex relations because the sides never meet.

The orientation-preserving subgroup consists of even words. Every pair $J_i{J}_j$ can be expressed in $A=J_2{J}_1$ and $B=J_3{J}_2$: the other nontrivial pairs are their inverses and $J_3{J}_1=BA$, $J_1{J}_3=A^{-1}{B}^{-1}$. Thus it is exactly $G$. A fundamental set is $U\cup J_{2}U$, with one representative chosen on each paired side; its interior is a fundamental domain. To see there are no relations between $A,B$, the reflection-word tree has vertices the reduced words and edges joining $w$ to $w{J}_i$. The even subgroup acts freely on this tree; its quotient has two vertices joined by three edges. Choosing the $J_2$ edge as a spanning tree leaves two independent loops, represented by $A$ and $B$. Reduced loops in a graph form a free group, soThe quotient is the double of the central three-sided region along its geodesic sides: an orientable pair of pants, topologically a sphere with three points removed. Its three ends are hyperbolic funnels, not cusps, because the sides are ultraparallel and the peripheral transformations have positive translation lengths. The compact convex core has three geodesic boundary circles; the complete quotient itself has no boundary.