\begin{tikzcd}[column sep=small, row sep=small]
\widetilde{S} \arrow[rr] & & S \\
\widetilde{G} = \mathrm{Sl}(2,\mathbb{R}) \arrow[rr] \arrow[d, no head] & &
G = \mathrm{GP}(2,\mathbb{R}) = \widetilde{G}/(+1,-1)
\arrow[d, no head] \\
\widetilde{\Gamma} \arrow[rr, "\sim"] & & \Gamma
\end{tikzcd}

\begin{tikzcd}
V \\
D \arrow[d] \\
X
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& H^{*}(B_{G},\mathbb{Z}) \arrow[rr] \arrow[dl] \arrow[d, "\wr"] & &
H^{*}(B_{\Gamma},\mathbb{Z}) \simeq H^{*}(X,\mathbb{Z}) \\
H^{*}(B_{\widetilde{G}},\mathbb{Z}) \arrow[d, "\wr"] &
H^{*}(B_{S},\mathbb{Z}) \simeq \mathbb{Z}[\gamma_{1}] \arrow[dl] & & \gamma_{1}^{X} \\
H^{*}(B_{\widetilde{S}},\mathbb{Z}) \simeq \mathbb{Z}[c_{1}] & & &
\end{tikzcd}

\begin{tikzcd}
B_{\Gamma} = X \arrow[r] \arrow[dr, dashed] & B_{G} \\
& B_{\widetilde{G}} \arrow[u]
\end{tikzcd}

\begin{tikzcd}
B_{\Gamma'} = X' \arrow[r, "p"] & B_{\Gamma} = X \arrow[r] & B_{G}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\mathrm{Br}(\bar{Y}) \arrow[r] \arrow[d, "\alpha"] &
\mathrm{Br}(\bar{X}) \arrow[r] \arrow[d, "\beta"] &
H^{1}(\bar{Y},\bar{P}) \arrow[r] \arrow[d, "\gamma"] &
H^{3}(\bar{Y},\underline{\mathbb{G}}_{m}) \arrow[r] \arrow[d, "\alpha'"] &
H^{3}(\bar{X},\underline{\mathbb{G}}_{m}) \arrow[d, "\beta'"] \\
\mathrm{Br}(y) \arrow[r] & \mathrm{Br}(X_{y}) \arrow[r] &
H^{1}(y,P_{0}) \arrow[r] & H^{3}(y,\underline{\mathbb{G}}_{m}) \arrow[r] &
H^{3}(X_{y},\underline{\mathbb{G}}_{m})
\end{tikzcd}

\begin{tikzcd}
(\mathrm{i}) \arrow[r, Rightarrow] \arrow[d, Rightarrow] &
(\mathrm{i\ bis}) \arrow[dl, Rightarrow] \\
(\mathrm{ii\ bis}) \arrow[r, leftrightarrow] \arrow[d, Rightarrow] &
(\mathrm{iii}) \\
(\mathrm{ii}) &
\end{tikzcd}