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## infinity of Markoff triples: $z=1$
$\begin{pmatrix} 3 & -1 \\ 1 & 0 \end{pmatrix}$
is an automorph of
$$x^2 + y^2 - 3x y.$$
So $( v_n,v_{n+1},1)$ is a Markoff triple where
$\begin{pmatrix} x \\ y \end{pmatrix}= \begin{pmatrix}v_{n+1} \\ v_n \end{pmatrix} = \begin{pmatrix} 3 & -1 \\ 1 & 0 \end{pmatrix}^n \begin{pmatrix}1 \\ 1 \end{pmatrix}$
* snake graph and its perfect matchings
* "lengths" that verify a Ptolemy inequality
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## Group actions
<!-1- _transition: glow -1->
$\mathbb{Q}\cup \infty \subset$ circle/projective line
* $(a,c)\text{ primitive } \mapsto a/c \in \mathbb{Q}\cup \infty$
* $\begin{pmatrix} a & d \\ c & d \end{pmatrix} \mapsto$ arc joining $(a/c, b/d)$
* $(a/b, c/d)$ are Farey neighbors iff $|ad - bc | = 1$
* obvious transitive $SL(2,\mathbb{Z})$ action on Farey neighbors
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[source](https://www.mathi.uni-heidelberg.de/~pozzetti/trees/4.pdf)
* involution $(x^-,y,z) \mapsto (x^+, y,z) = (3yz - x^-, z,y)$
* elliptic involution swaps triangles fixes midpoint of diagonal
#### Exo
- Nielsen move $\rightarrow$ Vieta flip
- $tr\,ab + tr\,ab^{-1} = (tr\,a) (tr\,b)$
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### Simple representatives in homology
$\phi : \mathbb{Z}*\mathbb{Z} \rightarrow \mathbb{Z}^2 \simeq
H^1(T,\mathbb{Z})$.
abelianizing homomorphism.
- $\phi$ takes generators of $\mathbb{Z}*\mathbb{Z}$ to generators of $\mathbb{Z}^2$.
- $(p,q) \in \mathbb{Z}^2$ generator $\Leftrightarrow p,q$ coprime.
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- $m_{p/q} = \frac13 tr \hat{\rho}( \gamma_{p/q})$