Anosov theorem for coincidences on special solvmanifolds of type $(\mathrm {R})$

Abstract:

Suppose that $S$ and $S’$ are simply connected solvable Lie groups of type $(\mathrm {R})$ with the same dimension. We show that the Lefschetz coincidence numbers of maps $f,g:\Gamma \backslash S\to \Gamma ’\backslash S’$ between special solvmanifolds $\Gamma \backslash S\to \Gamma ’\backslash S’$ can be computed algebraically as follows: \[ L(f,g) = \det (G_* - F_*), \] where $F_*,G_*$ are the matrices, with respect to any preferred bases, of morphisms of Lie algebras induced by $f$ and $g$. This generalizes a recent result by S. W. Kim and J. B. Lee to special solvmanifolds of type (R). Moreover, we can drop the dimension match condition imposed in the latter result.