do invertible upper triangular matrices have infinite index in $\text{GL}_2(\mathbf{C})$ I have been trying to understand this question from Mathoverflow:
https://mathoverflow.net/questions/34044/group-cannot-be-the-union-of-conjugates
The discussion is interesting, but the specific example is a bit beyond me.  Could someone provide a proof that invertible upper triangular matrices have infinite index in $\text{GL}_2(\mathbf{C})$?
 A: Suppose $H,K\le G$ are subgroups. Since $[G:K]\ge [HK:K]=[H:H\cap K]$, to prove that $[G:K]$ is infinite it suffices to prove $[H:H\cap K]$ is infinite. 
Can you find an infnite subgroup $H\le\mathrm{GL}_2(\mathbb{C})$ for which $H\cap U$ is finite? (Where $U$ stands for the subgroup of upper diagonal matrices.)
A: Turning my comment to an answer per Joe's request:
$\mbox{GL}_2(\mathbb{C})$ is an open 4-(complex)-dimensional submanifold (8-real-dimensions) of $\mbox{Mat}_{2\times 2}(\mathbb{C}) \cong \mathbb{C}^4$ (say $x_{11},x_{12},x_{21},x_{22}$ are complex coordinates) since it is the complement of the closed zero locus of the determinant function $x_{11}x_{22} - x_{21}x_{12}$.
The subgroup $U$ of invertible upper triangular matrices is a hyperplane section (namely, $\{x_{21} = 0\}\cap \mbox{GL}_2(\mathbb{C})$) and hence is a closed 3-(complex)-dimensional submanifold (6-real-dimensions). Importantly, we can think of it as a 6-real-dimensional Lie group.
$U$ acts on $\mbox{GL}_2(\mathbb{C})$ by left multiplication. The action (call it $\rho$) is smooth (the coordinate functions are polynomial), proper (it acts by diffeomorphisms which are of course proper) and free (all left-multiplication group actions are free). Smoothness, properness and freeness of the action are sufficient for the quotient by the action to also be a manifold (see here).
By a dimension count, the resulting manifold $M := \mbox{GL}_2(\mathbb{C})/\rho$ is 1-complex-dimensional (2-real-dimensions). It's points are in bijection with the orbits of the action $\rho$ but these orbits are exactly the left cosets of $U$. Having positive dimension means $M$ is in particular infinite. So the index of $U$ in $\mbox{GL}_2(\mathbb{C})$ is also infinite.
