= Tangent-disk Schottky group
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A <Tangent-disk Schottky group> uses side-pairing disks with disjoint interiors but permits boundaries to meet. This convention must be distinguished from the strictly separated, purely loxodromic <Schottky group> convention. For example, the maps $A(z)=z+2$ and $B(z)=z/(iz+1)$ pair the half-plane disks $\Re z<-1,\Re z>1$ and the disks $|z-i|<1,|z+i|<1$. The <ping-pong lemma> gives a free discrete group. With $C=ABA^{-1}B^{-1}$, the subgroup generated by $B,C$ is conjugate by $M(z)=iz/(2-z)$ to the principal level-two modular group, generated by $\begin{pmatrix}1&0\\2&1\end{pmatrix}$ and $\begin{pmatrix}1&2\\0&1\end{pmatrix}$. Thus its <Kleinian limit set> contains the circle $|z-1|=1$ and its translate $|z-3|=1$. Its <Hausdorff dimension> is at least one and it cannot be <homeomorphic> to a single circle.
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