Anosov theorem for coincidences on nilmanifolds
Volume 185 / 2005
Fundamenta Mathematicae 185 (2005), 247-259
MSC: 55M20, 54H25, 57S30.
DOI: 10.4064/fm185-3-3
Abstract
Suppose that $L,L'$ are simply connected nilpotent Lie groups such that the groups $\gamma_i(L)$ and $\gamma_i(L')$ in their lower central series have the same dimension. We show that the Nielsen and Lefschetz coincidence numbers of maps $f,g : {\mit\Gamma}\backslash L\to {\mit\Gamma}'\backslash L'$ between nilmanifolds ${\mit\Gamma}\backslash L$ and ${\mit\Gamma}'\backslash L'$ can be computed algebraically as follows: $$ L(f,g)=\det(G_*-F_*),\quad N(f,g)=\vert L(f,g)\vert, $$ where $F_*, G_*$ are the matrices, with respect to any preferred bases on the uniform lattices ${\mit\Gamma}$ and ${\mit\Gamma}'$, of the homomorphisms between the Lie algebras $\mathfrak{L}, \mathfrak{L}'$ of $L, L'$ induced by $f,g$.
