An analytic approach to $P$-adic diffeomorphism group and Teichm\"{u}ller theory

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