\documentclass[tikz]{standalone}

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

\begin{document}

\begin{tikzpicture}[scale = 1.3]

%diagonals + diagonal labels
\draw[gray,thick,dashed] (0,1) to node[midway,below,yshift=3pt,xshift=-3pt,scale=0.75]{ $x_\alpha$} (1,0);
\draw[gray,thick,dashed] (-0.7,1.7) to node[midway,below,scale=0.75,xshift=-2pt,yshift=2pt]{$x_\rho$} (1,1);
\draw[gray,thick,dashed] (0,2.4) to node[midway,below,scale=0.75,xshift=-2pt,yshift=2pt]{$x_\rho$} (1.7,1.7);
\draw[gray,thick,dashed] (1,2.4) to node[midway,below,scale=0.75,yshift=2pt]{ $x_\alpha$}  (2.4,2.4);

%snake graph outline
\draw[thick] (-0.7,1.7) to node[midway,above,xshift=-15pt, yshift=-1pt,scale=0.5]{$U_0(\lambda_\infty) x_\rho$} (1.7,1.7);
\draw[ultra thick,white] (0,2.4) to (1,1);
\draw[thick] (0,2.4) to node[midway,xshift=16pt,yshift=-7pt,scale=0.5]{$U_2(\lambda_\infty) x_\rho$} (1,1);
\draw[thick] (0,0) to (1,0) to node[midway,right,xshift=-2pt,scale=0.75]{$x_\rho$} (1,1) to node[midway,below,xshift=3pt,yshift=2pt,scale=0.75]{$x_\beta$} (0,1) to (0,0);
\draw[thick] (0,1) to node[midway,below,,xshift=-1pt,scale=0.75]{$x_\alpha$} (-0.7,1.7) to node[midway,above,xshift=-9pt,scale=0.5]{$U_1(\lambda_\infty)x_\rho$} (0,2.4) to node[midway,above,scale=0.75,yshift=-2pt]{$x_\alpha$} (1,2.4) to node[midway,right,xshift=-2pt,yshift=1pt,scale=0.75]{$x_\beta$}(1.7,1.7) to node[midway,below,scale=0.5,xshift=18pt]{$U_1(\lambda_\infty)x_\rho$} (1,1);
\draw[thick] (1,2.4) to (1.7,3.1) to (2.4,2.4) to node[midway,right,yshift=-3pt,xshift=-2pt,scale=0.75]{$x_\rho$} (1.7,1.7);

\end{tikzpicture}

\end{document}