For each prime number $p$ and number field $k$, there exists at least one extension $k_{\infty}/k$ with Galois group isomorphic to $\mathbb{Z}_p$, the cyclotomic $\mathbb{Z}_p$-extension. If $k_p/k$ is the maximal abelian $p$-unramified pro-$p$ extension of $k$, and if $G_p = \mathrm{Gal}(k_p/k)$, then $\mathrm{Gal}(k_{\infty}/k)$ is a quotient of $G_p$. The group $\mathbb{Z}_p$ is a free object within the theory of pro-p groups, so in fact there is a (non-unique) decomposition $G_p = \mathbb{Z}_p \times \Gamma$ for some pro-p group $\Gamma$.

When $k$ is totally real, it is known that the $p$-adic regulator of $k$ is nonzero if and only if $\Gamma$ is finite. My question is, when the $p$-adic regulator is nonzero, what is known about the relationship between its p-valuation and the group $\Gamma$? In particular, is there a relationship between this valuation and the exponent of $\Gamma$?