\documentclass[tikz]{standalone}

\usetikzlibrary{decorations.pathreplacing,decorations.markings}
\usetikzlibrary{patterns}
\begin{document}

\begin{tikzpicture}
%draw exterior boundaries
\draw[out=-30,in=-150] (0,0) to (1,0);
\draw[out=70,in=-30] (1,0) to (0.5,1.5);
\draw[out=-150,in=120] (0.5,1.5) to (0,0);

%draw vertices
\draw[fill=black] (0,0) circle[radius=2pt];
\draw[fill=black] (1,0) circle[radius=2pt];
\draw[fill=black] (0.5,1.5) circle[radius=2pt];

% draw orbifold point
\draw[thick] (0.5,0.7) node {$\mathbf{\times}$};

%draw ordinary arc
\draw[out=-10,in=-40,looseness=1.5] (0,0) to (0.5,1.5);

%draw pending arc
\draw[out=0,in=-60,looseness=0.75] (0,0) to (0.7,0.9);
\draw[out=120,in=90,looseness=0.75] (0.7,0.9) to (0,0);

% draw elementary lamination
\draw[out=-10,in=-40,looseness=1.25, densely dashed, red] (0.2,-0.15) to (0.7,1.45);

%edge labels
\node[scale=0.5] at (0.58,1.3) {$\tau_i$};
\node[scale=0.5,red] at (1,1.3) {$L_i$};

\end{tikzpicture}

\end{document}