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 $\widehat{G T}^{1}$ be the subgroup of the Grothendieck-Teichmüller group having $\lambda$-component equal to 1 . We define a subgroup $\Lambda$ of $\widehat{G T^{1}}$ by adding one additional defining relation to the definition of $\widehat{G T}^{1}$, and show that $\Lambda$ acts on the tower of profinite mapping class groups $\hat{\Gamma}_{g, n}^{m}$ for all $g, n, m \geqq 0$, respecting all the natural arrows $\hat{\Gamma}_{g^{\prime}, n^{\prime}}^{m^{\prime}} \rightarrow \hat{\Gamma}_{g, n}^{m}$ coming from cutting out a topological surface of genus $g^{\prime}$ with $n^{\prime}$ punctures and $m^{\prime}$ boundary components inside one of genus $g$ with $n$ punctures and $m$ boundary components. The proof that these homomorphisms are respected is an easy consequence of a certain local inertia conjugation property of the action of $\Lambda$.
What is the meaning of local inertia conjugation property in this paper? What is the meaning of conjugation?
