Multidimensional Gauss Reduction Theory for conjugacy classes of SL(n,Z)

In this paper we describe the set of conjugacy classes in the group SL(n,Z). We expand geometric Gauss Reduction Theory that solves the problem for SL(2,Z) to the multidimensional case. Further we find complete invariant of classes in terms of multidimensional Klein-Voronoi continued fractions, where $\varsigma$-reduce Hessenberg matrices play the role of reduced matrices.