  The extraordinary properties and the novel applications of black phosphorene
induce the research interest on the monolayer group-IV monochalcogenides. Here
using the first-principles calculations, we systematically investigate the
electronic, transport and optical properties of monolayer $\alpha-$ and
$\beta-$GeSe, the latter of which was recently experimentally realized. We
found that, monolayer $\alpha-$GeSe is a semiconductor with direct band gap of
1.6 eV, and $\beta-$GeSe displays indirect semiconductor with the gap of 2.47
eV, respectively. For monolayer $\beta-$GeSe, the electronic/hole transport is
anisotropic with an extremely high electron mobility of 7.84
$\times10^4$$cm^2/V\cdot {s}$ along the zigzag direction, comparable to that of
black phosphorene. Furthermore, for $\beta-$GeSe, robust band gaps nearly
disregarding the applied tensile strain along the zigzag direction is observed.
Both monolayer $\alpha-$ and $\beta-$GeSe exhibit anisotropic optical
absorption in the visible spectrum.
