Geodesic
CLUSTER STRUCTURES ON HIGHER TEICHMULLER SPACES FOR CLASSICAL GROUPS
Forum of Mathematics, Sigma, Tome 7 (2019)

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Let $S$ be a surface, $G$ a simply connected classical group, and $G^{\prime }$ the associated adjoint form of the group. We show that the moduli spaces of framed local systems ${\mathcal{X}}_{G^{\prime },S}$ and ${\mathcal{A}}_{G,S}$, which were constructed by Fock and Goncharov [‘Moduli spaces of local systems and higher Teichmuller theory’, Publ. Math. Inst. Hautes Études Sci.103 (2006), 1–212], have the structure of cluster varieties, and thus together form a cluster ensemble. This simplifies some of the proofs in that paper, and also allows one to quantize higher Teichmuller space, which was previously only possible when $G$ was of type $A$. 
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     author = {IAN LE},
     title = {CLUSTER {STRUCTURES} {ON} {HIGHER} {TEICHMULLER} {SPACES} {FOR} {CLASSICAL} {GROUPS}},
     journal = {Forum of Mathematics, Sigma},
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     doi = {10.1017/fms.2019.5},
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IAN LE. CLUSTER STRUCTURES ON HIGHER TEICHMULLER SPACES FOR CLASSICAL GROUPS. Forum of Mathematics, Sigma, Tome 7 (2019). doi: 10.1017/fms.2019.5

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