Depth filtration of the Nottingham group (${\mathcal{N}}_m=\{f:f(t)\equiv t(\mathrm{mod}{t}^{m+1})\}$)

Put ${\mathcal{N}}_m=\{f:f(t)\equiv t(\mathrm{mod}{t}^{m+1})\}$ for $m\ge 1$. These are open normal subgroups and ${\mathcal{N}}_m/{\mathcal{N}}_{m+1}\cong ({\mathbb{F}}_p,+)$ via the coefficient of $t^{m+1}$. Thus $[\mathcal{N}:{\mathcal{N}}_m]=p^{m-1}$. The depth of a nonidentity series is ${\mathrm{ord}}_t(f(t)-t)-1$. If $f=t+a{t}^{r+1}+\cdots$ and $g=t+b{t}^{s+1}+\cdots$, expansion of substitution givesHere $[f,g]=f^{-1}\circ g^{-1}\circ f\circ g$. This formula controls commutator subgroups and normal subgroups.

For odd $p$, group commutator collection in the depth filtration gives $[{\mathcal{N}}_m,{\mathcal{N}}_m]={\mathcal{N}}_{2m+1}$. Every degree $k\ge 2m+1$ occurs as a leading group commutator degree by choosing $r,s\ge m$, $r+s=k$, and $r\not\equiv s(\mathrm{mod}p)$; changing $r$ by one changes $r-s$ by two. Closed successive coefficient correction then fills the tail. Therefore ${\mathcal{N}}^{(n)}={\mathcal{N}}_{2^{n+1}-1}$ and ${\log }_p[\mathcal{N}:{\mathcal{N}}^{(n)}]=2^{n+1}-2$. In characteristic two the sharper collection theorem gives ${\log }_2[\mathcal{N}:{\mathcal{N}}^{(n)}]\le C{\kappa }^n$ for a constant $C$, with $\kappa =(1+\sqrt{17})/2<3$. In both cases the exponential base is strictly smaller than three.