I'm considering the possible combinations of these properties for an isometry group: "finite", "finitely generated", "discrete".
Obviously, a finite group is necessarily finitely generated and discrete. The remaining possibilities are for infinite groups.
The group may be both finitely generated and discrete. Example: The group of translations in $\mathbb R$ by integers, is generated by a single translation ($x\mapsto x+1$).
The group may be finitely generated but not discrete. Example: The group of rotations in $\mathbb R^2$ by multiples of an irrational angle (like $1$ radian $= 1/(2\pi)$ cycles), maps a single point to a set of points on a circle which come arbitrarily close to the original point.
The group may be not discrete nor finitely generated. Example: The group of all translations in $\mathbb R$.
Can the group be discrete but not finitely generated? If we use $\mathbb R^\infty$ (the space of sequences with finitely many non-zero terms), the group generated by reflections along the basis vectors $\{(1,0,0,0,\cdots),(0,1,0,0,\cdots),(0,0,1,0,\cdots),\cdots\}$ seems to work. But I don't want to use an infinite-dimensional space.
I think I've proven that a group of finite-dimensional translations, if discrete, must be finitely generated. The proof also works for rotations in $\mathbb R^2$. I could try to generalize it to rotations in higher dimensions (of Euclidean space), but there's still hyperbolic space.
I found this about Fuchsian groups, which are discrete isometry groups for the hyperbolic plane:
There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated…
Is that a necessary assumption? Are there Fuchsian groups that aren't finitely generated?
Best Answer
It is easy to find a discrete group of isometries of the hyperbolic plane which is isomorphic to $F_2$, the free group with two generators. The generators could be e.g. "go 10 units" and "rotate 90 degrees right, go 10 units, rotate 90 degrees left". (You can see an interactive demo of this here -- press 'g' to mark places, the distance here is smaller than 10 units, but large enough that you will never reach the starting point unless by retracing your path, thus proving that the group is indeed free.)
And $F_2$ has subgroups which are not finitely generated. Subgroups of finitely generated groups are not necessarily finitely generated