$\newcommand{\GL}{\operatorname{GL}}$

Let $H_{>k}$ be the space of real $d \times d$ matrices of rank bigger than $k$, for some fixed $k$. $H_{>k}$ is an open connected submanifold of $ \mathbb{R}^{d^2}$.

Question:Does $H_{>k}$ admit a transitive Lie group action? (by a "Lie group" I mean a finite dimeniosnal one).

A manifold that admits a transitive Lie group action has finitely-generated homotopy groups $\pi_n$ for $n\ge 2$-and $H_{>k}$ satisfies this necessary condition.

There is an action on $H_{>k}$ by $\GL(n) \times \GL(n)$, given by $$ (A,B) \cdot C=ACB^{-1}. $$

The orbits correspond to the different ranks. Even if there is a transitive action, I don't expect it to be as "natural".

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