# Friday April 17th

Let $X$ be locally compact and $\Gamma \actson X$ by homeomorphisms.
Recall that $\Gamma$ acts properly discontinuously at $x\in X$ iff there exists a neighborhood $N_x$ such that $\theset{\gamma\in \Gamma \suchthat \gamma\cdot N_x \cdot N_x \neq \emptyset}$ is finite.
Equivalently, th e orbit $\Gamma\cdot x$ is discrete in $X$ and the stabilizer $\Gamma^x$ is finite.
Moreover $\Gamma$ acts freely iff $\Gamma^x$ is trivial for all $x$.

Proposition
: For such $X, \Gamma$ as above, the map $X \to \Gamma/X$ is a galois covering iff $\Gamma$ acts freely and properly discontinuously.

> Note: Galois covering here is just the usual notion involving deck transformations.

This requires $\Gamma$ to be discrete, otherwise a sequence converging to the identity would have infinitely many neighborhoods converging on a neighborhood of the identity.

A *Fuchsian* group is a discrete subgroup of automorphisms of $\HH \subset \CC$, i.e. $\Gamma \subset \aut_\CC (\HH) = \PSL(2, \RR)$.

Fact
:   Every Fuchsian group acts properly discontinuously on all of $\HH$.
    Thus the discreteness which is necessary is also sufficient in this case.
    Then $\HH \to \Gamma \textbackslash \HH$ is a covering map and the action is free $\iff \Gamma$ has no elements of finite order.

Thus the complex structure (sheaf of holomorphic functions?) pushes forward to the quotient, giving it the structure of a Riemann surface.

Definition
: A *Kleinian* group is a discrete subgroup of $\PSL(2, \CC) = \PGL(2, \CC) = \aut_\CC(\PP^1)$.

Note that $\HH$ carries a hyperbolic structure, and this group precisely preserves it.

Define the regular set as $\ell(\Gamma) = \theset{x\in \PP^1 \suchthat  x \text{ acts freely and PDC}}$

Fact
: A Kleinian group is conjugate in $\PSL(2, \CC)$ to a Fuschian group if it stabilizes a circle in $\PP^1(\CC)$.

Exercise
:   Let $\Gamma = \gens{[q, 0; 0, 1]} \subset \PSL(2, \CC)$ and $\Lambda(\Gamma) = \PP^1\setminus \theset{\vector 0}$.
    Show $\Lambda(\Gamma)$ is open but could be empty, has either 0,1,2, or infinitely many connected components.

Exercise
:   Show that for $\Gamma = \PSL(2, \ZZ)$, $\Lambda(\Gamma) = \HH^+ \union \HH^-$.

If $U(\Gamma) \neq \emptyset$, then $\Lambda(\Gamma) \to \Lambda(\Gamma)/\Gamma$ is a galois covering that endows the quotient with the structure of a Riemann surface.

For $G$ a group and $X$ a set with a map $M: G\cross X \to X$, we say $X$ is a $G\dash$set.
We can make a category from this: given $(X, \mu_x)$ and $(Y, \mu_y)$, a $G\dash$map is given by a diagram

\begin{tikzcd}
G\cross X \ar[r, "\mu_x"] \ar[d, "\id \cross f"] & X \ar[d, "f"] \\
G\cross Y \ar[r, "\mu_y"]                        & Y
\end{tikzcd}


The set $X$ is transitive if for all $x, y\in X$, there is a $g\in G$ such that $gx = y$.
$X$ is *simply transitive* if this $g$ is unique.

Exercise
:   $G\cross G \mapsvia{\mu} G$ given by group multiplication is simply transitive.
    Conversely, any simply transitive $G\dash$set $X$ is isomorphic to $G$.
    Choosing $p_0 \in X$, consider the map $\phi: X\to G$ where $p \mapsto p - p_0$.

This is a torsor, i.e. a principal homogeneous space?

Example:
ODEs, the solutions for the inhomogeneous equation are torsors for the homogeneous solutions (?).


Let $k$ be a field and $G/k$ a commutative algebraic group.
A torsor (or principal homogeneous space) under $G$ is a nonempty $k\dash$variety $X/A$ equipped with a morphism $\mu: G\cross X \to X$ such that the following holds.

1st try: define $\mu(\bar k): G(\bar k) \cross X(\bar k) \to X(\bar k)$ to be a simply transitive action.
This is a good definition in characteristic zero.
In general, we have an isomorphism 
\begin{align*}
G\cross X &\to X
(g, x) &\mapsto (x, gx)
.\end{align*}
This is trivial iff $(X, \mu) \cong (G, \cdot)$ with its group structure.

Claim
:   $X$ is trivial $\iff X(k) \neq \emptyset$

Proof
:   $\implies: X \cong_K G$ and $G(k) \neq \emptyset$, so $X(k)\neq \emptyset$.
    \
    $\impliedby:$ Let $p_0 \in X(k)$, then the map 
    \begin{align*}
    \mu(\wait, p_0): G \to X
    \end{align*}
    is defined over $k$ and is a bijection on $X(\bar k)$, and thus an isomorphism in characteristic zero.

This implies that every $X$ under $G$ is a twisted form of $G$, i.e. $X/\bar k \cong G/\bar k$.


There's a group law on the torsors called the Baer sum, and a group structure yielding the Weil-Chevallet group.
Every torsor will be given by a galois cohomological cocycle.