Emergency push!

src/dyn/top/stone-cech.tex

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+\section{$G \curvearrowright \beta G$}
+\label{section:stone-cech-flow}
+
+\begin{definition}[Greatest Ambit]
+\label{definition:stone-cech-flow}
+    Let $G$ be a discrete group, and
+    \[
+    \beta L: G \to \text{Aut}(\beta G) \quad \beta L_g(\fU) = g\fU
+    \]
+
+    then
+    \begin{enumerate}
+        \item $(\beta L: G \curvearrowright \beta G, 1_G)$ is an ambit.  
+        \item[(U)] For any ambit $(G \curvearrowright X, x_0)$, there exists a unique morphism of pointed $G$-flows $\pi: (\beta G, 1_G) \to (X, x_0)$. Moreover, $\pi$ is surjective. 
+    \end{enumerate}
+    
+    Any ambit $(Y, y_0)$ satisfying (U) is the \textbf{greatest $G$-ambit}. 
+\end{definition}
+\begin{proof}
+    (1): Let
+    \[
+    L: G \to \text{Perm}(G) \quad L_g(h) = gh
+    \]
+
+    be the left translation action on itself. For each $g \in G$, the functoriality of the \hyperref[Stone-Čech compactification]{definition:stone-cech} implies that $L_g$ extends into a continuous map $\beta L_g: \beta G \to \beta G$. By uniqueness of continuous extensions, $\beta L_g(\fU) = g\fU$ for all $\fU \in \beta G$. In particular, $\text{Id}_{\beta G} = \beta L_{e}$. 
+    
+    Moreover, for any $g, h \in G$, $\beta L_{gh} = \beta L_{g} \circ \beta L_{h}$, so $g \mapsto L_g$ is a homomorphism. Finally, for any $g \in G$, 
+    \[
+    \beta L_{g^{-1}} \circ \beta L_{g} = \beta L_{1_G} = \beta L_{g} \circ \beta L_{g^{-1}}
+    \]
+    
+    and $\beta L_g \in \text{Aut}(\beta G)$. 
+
+    Since $\bracsn{L_g1_G|g \in G} = G$ is dense in $\beta G$, $(G \curvearrowright \beta G, 1_G)$ is an ambit. 
+
+    (2): For each $g \in G$, let $\pi_0(g) = gx_0$, then $\pi_0 \in C(G; X)$, $\pi_0(1_G) = x_0$, and the following diagram commutes
+    \[
+    \xymatrix{
+    G \ar@{->}[r]^{\pi_0} \ar@{->}[d]_{g} & X \ar@{->}[d]^{g} \\
+    G \ar@{->}[r]_{\pi_0} & X
+    }
+    \]
+
+    for all $g \in G$. By universal property of $\beta G$, $\pi_0$ admits a unique continuous extension $\pi \in C(\beta G; X)$. By uniqueness of continuous extensions, the following diagram commutes
+    \[
+    \xymatrix{
+    \beta G \ar@{->}[r]^{\pi} \ar@{->}[d]_{g} & X \ar@{->}[d]^{g} \\
+    \beta G \ar@{->}[r]_{\pi} & X
+    }
+    \]
+
+    for all $g \in G$. Finally, since $\beta G$ is compact and $X$ is Hausdorff, $\pi(\beta G)$ is \hyperref[also compact]{proposition:compact-extensions} and \hyperref[hence closed]{proposition:compact-closed}. As $(X, x_0)$ is an ambit and $\pi(\beta G) \supset Gx_0$, $\pi(\beta G) \supset \ol{Gx_0} = X$. 
+\end{proof}
+