Let $SL(n, \mathbb{R})$ be the group of $n \times n$ invertible matrices of determinant $1$ in real numbers. Let $G:=SL(n, \mathbb{R}_{\geq 0})$ be its subgroup $\{M \in SL(n,\mathbb{R}) \mid M, M^{-1} \text{ both have non-negative entries}\}$. Is there any known results on this group? I am particular interest in how big (roughly) this group could be. I know $S_n \subset SL(n, \mathbb{R}_{\geq 0})$ (considering the permutation of a basis of a $n$- dimensional vector space), but I wish it could be much bigger than that.