Fundamental group of a component of $GL_n({\bf R})$

Question

Let $G$ be a component of $GL_n({\bf R})$ such that element has a positive determenant.

(1) Since it contains $SO(n)$, $\pi_1(SO(n))$ ? What is a fundamental group of $G$ ?

(2) It has a curvature bound ? That is to say, we can have bound $-1$ below or $-\infty$ ?

Answer 1

$(1)$ Yes, like $SO(n)$, the group $GL_+(n, \mathbb{R})$ is not simply connected for $n\ge 2$, but rather has a fundamental group isomorphic to $\mathbb{Z}$ for $n=2$ or $\mathbb{Z}/2\mathbb{Z}$ for $n>2$.