Finite symmetries for embeddings of genus $\geq 2$ surfaces in $\mathbb{R}^3$ Let $f : \Sigma \to \mathbb{R}^3$ be a genus $g \geq 2$ surface smoothly embedded in $\mathbb{R}^3$.  Let 
$$
G(f) = \{ \phi \in \text{Isom}(\mathbb{R}^3) : \phi(f(\Sigma)) = f(\Sigma)\}
$$ 
be the group of isometries of $\mathbb{R}^3$ that preserve $\Sigma$.  Is the order of $G(f)$ always finite?  If so is there a bound on the order of $G(f)$ (presumably in terms of $g$)?  

Benson-Tilsen's paper "Isometry Groups of Compact Riemann Surfaces" seems to have part of the answer. Be careful, though; it only deals with orientation-preserving isometries ($G^+ \neq G$), so the numbers there are half of what they could be. And it doesn't deal with embedding.
The important result, for the maximum possible size of the group for any surface with genus $g \geq 2$, is
$$8(g+1) \leq \max|G^+| \leq 84(g-1)$$
(The max itself is a bound on the group's size, and this is a bound on the max.)
