The Reidemeister numbers and the Hirsch ranks for preimage subgroups

초록

Let theta : gamma(1) -> gamma(2) be a group homomorphism and lambda subset of gamma(2) a subgroup. We consider the Reidemeister number R(theta, lambda) of theta at lambda, which generalizes the classical Reidemeister number. We study a 6-term exact sequence of preimage subgroups and Reidemeister sets induced from a commutative diagram between short exact sequences of groups. When the quotient groups are finite in the commutative diagram, we study an averaging formula for the generalized Reidemeister numbers. When gamma(1) and gamma(2) are finitely generated torsion-free nilpotent groups, we prove that R(theta, lambda) < infinity <--> rk(gamma(1)) - rk(theta(-1)(lambda)) = rk(gamma(2)) - rk(lambda) where rk(gamma) denotes the Hirsch rank of a finitely generated torsion-free nilpotent group gamma. This result is a generalization of the coincidence re-sult of Gon , calves (Topol Meth Nonlin Anal 12:375-386, 1998) to the preimage result. We extend this coincidence result of Gon , calves from finitely generated nilpotent groups to discrete cocompact subgroups of the three-dimensional solvable Lie group Sol. As an application, we de-rive an averaging formula of the Reidemeister coincidence numbers for homomorphisms between Bieberbach subgroups of Sol.

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