We give a practical criterion to determine whether a given pair of morphisms between almost-crystallographic groups has a finite Reidemeister coincidence number. As an application, we determine all two-and three-dimensional almost-crystallographic groups that have the R∞ property. We also show that for a pair of continuous maps between oriented infra-nilmanifolds of equal dimension, the Nielsen coincidence number equals the Reidemeister coincidence number when the latter is finite.Mathematics Subject Classification (2010). Primary 20E45; Secondary 55M20, 20H15.

In this paper, we investigate the finiteness of the Reidemeister number R(f ) of a selfmap f : M → M on an infra-nilmanifold M . We show that the Reidemeister number of an Anosov diffeomorphism on an infra-nilmanifold is always finite. A manifold M is said to have the R∞ property if R(f ) = ∞ for every homeomorphism f : M → M . We show that every non-orientable generalised Hantzsche-Wendt manifold has the R∞ property. For an orientable Hantzsche-Wendt manifold M , we formulate a criterion, in terms of an associated graph, for M to have the R∞ property.

Abstract. Suppose that S and S are simply connected solvable Lie groups of type (R) with the same dimension. We show that the Lefschetz coincidence numbers of maps f, g : Γ\S → Γ \S between special solvmanifolds Γ\S → Γ \S can be computed algebraically as follows: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.

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We prove practical formulas for the Reidemeister coincidence number, the Lefschetz coincidence number and the Nielsen coincidence number of continuous maps between oriented infra-nilmanifolds of equal dimension. In order to obtain these formulas, we use the averaging formulas for the Lefschetz coincidence number and for the Nielsen coincidence number and we develop an averaging formula for the Reidemeister coincidence number. We also give a simple proof of the averaging formula for the Lefschetz coincidence number. Mathematics Subject Classification 2000: 55M20; 57S30.