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12.70 (1992)

Solved

Let $p$ be a prime number, $F$ a free pro-$p$-group of finite rank, and $\Delta \neq 1$ an automorphism of $F$ whose order is a power of $p$. Is the rank of $\text{Fix}_F(\Delta) = \{x \in F \mid \Delta(x) = x\}$ finite? If the order of $\Delta$ is prime to $p$, then $\text{Fix}_F(\Delta)$ has infinite rank (W. Herfort, L. Ribes, Proc. Amer. Math. Soc., 108 (1990), 287–295).

Contributor: L. Ribes

Progress

Yes, it is finite (C. Scheiderer, Proc. Amer. Math. Soc., 127, no. 3 (1999), 695–700). Moreover, in (W. N. Herfort, L. Ribes, P. A. Zalesskii, Forum Math., 11, no. 1 (1999), 49–61) it is proved that if $P$ is a finite $p$-group of automorphisms of a free pro-$p$ group $F$ of arbitrary rank, then the group of fixed points of $P$ is a free factor of $F$ and the latter result has been extended in (P. A. Zalesskii, J. Reine Angew. Math., 572 (2004), 97–110) to the situation of a free pro-$p$ group of arbitrary rank.

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