G acts effectively implies $ dim(G) \leq \frac{n(n+1)}{2} $

Question

Suppose that a Lie group $ G $ acts effectively on a manifold $ M $ with $ dim(M)=n $ then is it true that $$ dim(G) \leq \frac{n(n+1)}{2} $$ For the special case that $ G $ is compact this can be proven by induction on $ n $ because then we can equip $ M $ with a left $ G $ invariant metric so $ G $ acts by isometries and thus for each $ x $ the orbit $ \mathcal{O}_x \leq M $ has dimension at most $ n $ while the stabilizer $ G_x $ acts effectively on the unit sphere in the tangent space $ T_xM $ and $ G_x $ has dimension at most $ \frac{n(n-1)}{2} $. So $$ dim(G) = dim(\mathcal{O}_x) + dim(G_x) \leq n+ \frac{n(n-1)}{2}= \frac{n(n+1)}{2} $$

Answer 1

Let $M=\mathbb{R}$. Then there is an effective action by the $2$-dimensional group of affine maps $\{x\mapsto ax+b|a\in \mathbb{R}\backslash\{0\},\,\,b\in \mathbb{R}\}$.

Also another example, where $M$ is compact: $PSL_2(\mathbb{R})$ is $3$-dimensional and acts effectively on $S^1=\mathbb{R}P^1$ by Mobius transforms.