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What is the meaning of local inertia conjugation property?

In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030., we have: Abstract. Let $\widehat{G T}^{1}$ ...
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