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.



Subgroup of $SL(n,\mathbb{R})$ with positive entries
This is answered as Pietro says in <http://archive.numdam.org/ARCHIVE/CM/CM_1969__21_4/CM_1969__21_4_376_0/CM_1969__21_4_376_0.pdf>


In general semigroup theorists have heavily studied maximal subgroups of semigroups of nonnegative matrices. 


Again all maximal subgroups are isomorphic to the group of monomial matrices with nonnegative entries of appropriate rank. 


