$G_2$ as the symmetry group of a geometric object Is there a seven-dimensional geometric object whose full group of symmetries is isomorphic to the compact Lie group $G_2$, or does the same problem as with the special orthogonal groups occur?
 A: The OP doesn't say what is meant by a 'geometric object', so it's hard to give a definitive answer.  However, if one assumes that the geometric object is a smooth manifold $M^7$ and that the action is smooth, then there are a few things one can say:
First, the $\mathrm{G}_2$-stabilizer of any point $p\in M^7$ has to be a closed subgroup $H_p\subset \mathrm{G}_2$ which is therefore a compact Lie group of rank at most $2$ and dimension at least $7$.  The rank $1$ closed connected subgroups of $\mathrm{G}_2$ are either of dimension $1$ or $3$, so they are ruled out, so $H_p$ must be of rank $2$ and hence must contain a maximal torus of $\mathrm{G}_2$.  Looking at the root diagram, one sees that the only possibilities for such subgroups of dimension at least $7$ are conjugates of $\mathrm{SU}(3)$ (which is a maximal proper subgroup) and $\mathrm{G}_2$ itself.  Thus, the orbits of $\mathrm{G}_2$ on $M^7$ must be either $6$-spheres or points.
Obviously, there must be at least one $6$-sphere orbit, otherwise the action will be trivial.  We may as well discard the orbits that are points (if there are any), and we may as well restrict to a connected component of what remains, in which case $M^7$ is diffeomorphic to $\mathbb{R}\times S^6$, with the standard $\mathrm{G}_2$-action on $S^6$.  Let $t:\mathbb{R}\times S^6\to\mathbb{R}$ be projection on the first factor.
Now, $\mathrm{G_2}$ preserves a nondegenerate $2$-form $\omega$ on $S^6$
(and a compatible Riemannian metric $g$, which has constant sectional curvature $+1$). In fact, the $G_2$-invariant differential forms on $S^6$ are generated by $1$, $\omega$, $\mathrm{d}\omega$ (which is nowhere vanishing), and $\ast_g\mathrm{d}\omega$.  Meanwhile, the group of diffeomorphisms of $S^6$ that preserve $\omega$ is exactly $\mathrm{G}_2$.
Thus, for example, the group of diffeomorphisms of $\mathbb{R}\times S^6$ that preserve the $2$-form $t\,\omega$ is exactly $\mathrm{G}_2$.  Thus, if you count the manifold $\mathbb{R}\times S^6$ endowed with the $2$-form $t\,\omega$ as a geometric object, you have a $7$-dimensional geometric object whose symmetry group is $\mathrm{G}_2$.
