= 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> is
$$
W=\langle J_1,J_2,J_3\mid J_1^2=J_2^2=J_3^2=1\rangle
\cong C_2*C_2*C_2.
$$
The 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 $J_iJ_j$ can be expressed in $A=J_2J_1$ and $B=J_3J_2$: the other nontrivial pairs are their inverses and $J_3J_1=BA$, $J_1J_3=A^{-1}B^{-1}$. Thus it is exactly $G$. A fundamental set is \b[$U\cup J_2U$, 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 $wJ_i$. 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>, so
$$
\boxed{G\cong F_2\text{ freely on }A,B.}
$$
The quotient is the double of the central three-sided region along its geodesic sides: \b[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.
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