An analytic approach to $P$-adic diffeomorphism group and Teichmüller theory

We consider a specific class of infinite dimensional $p$-adic Lie groups, i.e., a sort of diffeomorphism groups on $p$-adic ball ${\mathrm{Diff}}^{\mathrm{an}}(B_{\epsilon })$. It turns out that this group has a natural logarithmic structure that leads to a $p$-adic version of Teichmüller theory on diffeomorphism groups, which also presents some remarkable hydrodynamic facets. We further apply this framework to Mochizuki’s $p$-adic Teichmüller theory and Inter-universal Teichmüller theory (IUT), and give a new reformulation of IUT as a Teichmüller theory on automorphisms of two-dimensional group schemes.