Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of invertible $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ there exists a matrix in $G$ that maps $u$ to $v$. Are there any non-trivial examples of such a $G$?



Transitive Semigroups of $2\times 2$ matrices
Here is a complete answer:


*Every semigroup* $S$ *of invertible* $2\times 2$\*-matrices which is transitive on\* $\mathbb R^2$ *is either conjugate to* $SO\_2(\mathbb R) \times \mathbb R^+$ *or* $SO\_2(\mathbb R) \times \mathbb R$ *or it is a product of* $SL\_2(\mathbb R)$ *and a multiplicative subgroup of* $\mathbb R$\*.\*


**Proof:** Let S be such a semigroup. Then the intersection $S\_0$ with $SL\_2(\mathbb R)$ is a subsemigroup of $SL\_2(\mathbb R)$. By a theorem of Hilgert and Hofmann (see their beautiful paper on "Old and new on $SL\_2$") there are only three choices for $S\_0$: Either $S\_0$ is all of $SL\_2(\mathbb R)$, a circle group or contained in a conjugate of the elements of $SL\_2(\mathbb R)$ with only positive entries. If $S\_0$ is a circle group, then $S$ will be conjugate to $SO\_2(\mathbb R) \times \mathbb R^+$ or $SO\_2(\mathbb R) \times \mathbb R$. If $S\_0$ happens to be all of $SL\_2(\mathbb R)$, then we have $SL\_2(\mathbb R)\subset S \subset GL\_2(\mathbb R)$, so $S$ is a product of $SL\_2(\mathbb R)$ and a multiplicative subgroup of $\mathbb R$. In the third case, we may assume that $S\_0$ is actually contained in the semigroup described above. Then $S\_0$ maps every vector with two positive entries to a vector with two posiitve entries, hence $S$ maps the upper right quadrant to a subset of itself and the lower left quadrant. In particular, $S$ cannot be transitive.


