$ \newcommand{\bF}{\mathbb{F}} \newcommand{\HH}{\mathbb{H}} \newcommand{\NN}{\mathbb{N}} \newcommand{\RR}{\mathbb{R}} \newcommand{\RP}{\mathbb{R}P} \newcommand{\PP}{\mathbb{P}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\calA}{\mathcal{A}} \newcommand{\calB}{\mathcal{B}} \newcommand{\calC}{\mathcal{C}} \newcommand{\calD}{\mathcal{D}} \newcommand{\calE}{\mathcal{E}} \newcommand{\calF}{\mathcal{F}} \newcommand{\calG}{\mathcal{G}} \newcommand{\calH}{\mathcal{H}} \newcommand{\calJ}{\mathcal{J}} \newcommand{\calL}{\mathcal{L}} \newcommand{\calM}{\mathcal{M}} \newcommand{\calN}{\mathcal{N}} \newcommand{\calO}{\mathcal{O}} \newcommand{\calP}{\mathcal{P}} \newcommand{\calQ}{\mathcal{Q}} \newcommand{\calS}{\mathcal{S}} \newcommand{\calT}{\mathcal{T}} \newcommand{\calV}{\mathcal{V}} \newcommand{\calX}{\mathcal{X}} \newcommand{\calZ}{\mathcal{Z}} \newcommand{\frakg}{\mathfrak{g}} \newcommand{\balpha}{{\boldsymbol \alpha}} \newcommand{\bbeta}{{\boldsymbol \beta}} \newcommand{\bgamma}{{\boldsymbol \gamma}} \newcommand{\bxi}{{\boldsymbol \xi}} \newcommand{\he}{\hat{e}} \newcommand{\hv}{\hat{v}} \newcommand{\hw}{\hat{w}} \newcommand{\oT}{\overline{T}} \newcommand{\oY}{\overline{Y}} \newcommand{\hDelta}{\widehat \Delta} \newcommand{\hA}{\widehat A} \newcommand{\hT}{\widehat T} \newcommand{\hU}{\widehat U} \newcommand{\hV}{\widehat V} \newcommand{\hX}{\widehat X} \newcommand{\hY}{\widehat Y} \newcommand{\hZ}{\widehat Z} \newcommand{\sfV}{\mathsf{V}} \newcommand{\talpha}{{\widetilde \alpha}} \newcommand{\tbeta}{{\widetilde \beta}} \newcommand{\tgamma}{{\widetilde \gamma}} \newcommand{\tsigma}{{\widetilde \sigma}} \newcommand{\txi}{{\widetilde \xi}} \newcommand{\tSigma}{{\widetilde \Sigma}} \newcommand{\ttau}{{\widetilde \tau}} \newcommand{\ta}{{\widetilde a}} \newcommand{\tb}{{\widetilde b}} \newcommand{\te}{{\widetilde e}} \newcommand{\tv}{{\widetilde v}} \newcommand{\tA}{{\widetilde A}} \newcommand{\tC}{{\widetilde C}} \newcommand{\tD}{{\widetilde D}} \newcommand{\tM}{{\widetilde M}} \newcommand{\tN}{{\widetilde N}} \newcommand{\tT}{{\widetilde T}} \newcommand{\tU}{{\widetilde U}} \newcommand{\tX}{{\widetilde X}} \newcommand{\tY}{{\widetilde Y}} % shortcuts \newcommand{\norm}[1]{\left\| {#1} \right\|} \newcommand{\abs}[1]{\left\lvert {#1} \right\rvert} \newcommand{\wght}[1]{\lvert {#1} \rvert} \newcommand{\I}[1]{\langle #1 \rangle} \newcommand{\bd}{\partial} \newcommand{\from}{\colon\thinspace} \newcommand{\inject}{\hookrightarrow} \newcommand{\inv}{^{-1}} \newcommand{\pmone}{\ensuremath{\{\pm 1\}}} \newcommand{\surject}{\twoheadrightarrow} \newcommand{\GP}[2]{\left( #1 \, . \, #2 \right)} \newcommand{\FN}{F_{N}} \newcommand{\dsym}{d^{\rm \, sym}} \newcommand{\margin}[1]{\marginpar{\scriptsize #1}} \newcommand{\dbtilde}[1]{\widetilde{\raisebox{0pt}[0.85\height]{$\widetilde{#1}$}}} \newcommand{\vsub}{\rotatebox[origin=c]{-90}{$\subseteq$}} % complexes \newcommand{\FF}{\mathrm{FF}} \newcommand{\ZF}{\mathcal{Z}\mathrm{F}} \newcommand{\coS}{\mathrm{CoS(\FN)}} $

Schottky Groups

it of four schottky circles under the action of a Schottky group

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