2 problems
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The collar lemma conjecture distinguishing higher rank Teichmüller theories
Collar lemma conjecture. The validity of the collar lemma distinguishes higher rank Teichmüller theories within the larger class of Anosov representations.
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Lee–Zhang conjecture on length-spectrum domination by Fuchsian representations
Lee–Zhang conjecture. For any Hitchin representation , there is a Fuchsian representation such that