# Non-commutative amoebas

Grigory Mikhalkin, Mikhail Shkolnikov

Grigory Mikhalkin, Mikhail Shkolnikov

**18/2/21**Published in : arXiv:2102.09324

The group of isometries of the hyperbolic 3-space is one of the simplest non-commutative complex Lie groups. Its quotient by the maximal compact subgroup naturally maps it back to the hyperbolic space. Each fiber of this map is diffeomorphic to the real projective 3-space.

The resulting map can be viewed as the simplest non-commutative counterpart of the amoeba map introduced, in the commutative setting, by Gelfand, Kapranov and Zelevinsky. The paper surveys basic properties of the non-commutative amoebas and compares them against their commutative counterparts.