\documentclass[tikz]{standalone}

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

\begin{tikzpicture}

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


%snake graph outline
\draw[thick] (0,1) to node[midway,left,scale=0.5,xshift=1pt, yshift = -8pt]{$U_1(\lambda_\infty) x_\rho$} (0,2);
\draw[thick, orange] (0,2) to node[midway,above,scale=0.75,xshift=-2pt]{$x_\alpha$}  (1,2);
\draw[thick] (1,2) to node[midway,right,scale=0.75,xshift=-2pt,yshift=8pt]{$x_\beta$}  (1,1);
\draw[thick, orange] (1,1) to node[midway,below,scale=0.5,xshift=1pt]{$U_{2}(\lambda_\infty) x_\rho$} (0,1);
%\draw[thick] (2,1) to node[midway,left,scale=0.5,yshift=-8pt,xshift=2pt]{$U_{k} x_\rho$}  (2,2) to node[midway,above,scale=0.75,xshift=-1pt]{$x_\beta$}  (3,2) to node[midway,right,scale=0.75,xshift=-1pt, yshift = 8pt]{$x_\alpha$} (3,1)  to node[midway,below,scale=0.5,yshift=-1pt]{$U_{k-1} x_\rho$}  (2,1);

\end{tikzpicture}

\end{document}