Noncommutative amoebas in the hyperbolic space and coamoebas in the 3-dimensional sphere

One of the simplest examples of noncommutative complex Lie groups is the group $SL_2(\mathbb C)$ of nonsingular $2\times 2$ matrices with nonzero determinant. From the topological viewpoint it can be identified with a cotangent bundle of a 3-dimensional sphere from. Matrix analogues of the logarithm of a norm and an argument define an amoeba-type map into the hyperbolic space and a coamoeba-type map into the 3-dimensional sphere. Properties of curves and surfaces under these mappings will be discussed in this talk. The talk is based on a joint work with Mikhail Shkolnikov.