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

Abstract

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.

The Nielsen and Reidemeister numbers of maps on infra-solvmanifolds of type (R)

“…Remark 3.7. In [11,Theorem 4.5] the Nielsen zeta function is expressed in terms of Lefschetz zeta functions L f (z) and L f + (z) via a table given by parity of p and n. The class of infra-solvmanifolds of type (R) contains and shares a lot of properties of the class of infra-nilmanifolds such as the averaging formula for Nielsen numbers, see [32,42]. Therefore, Theorem 1.2 and the statement about N f (z) in Theorem 3.6 can be generalized directly to the class of infra-solvmanifolds of type (R), see Remark in [11,Sec.…”

“…Since L(f n ) = det(I −D n * ) = 0 for all n > 0 by [32,Theorem 3.1], this would imply that the differential D * of D has no roots of unity. By Proposition 8.1, S must be nilpotent.…”

Nielsen zeta functions for maps on infra-nilmanifolds are rational

“…Remark. The class of infra-solvmanifolds of type (R) is a class of manifolds which contains the class of infra-nilmanifolds and which still shares a lot of the good (algebraic) properties of the class of infra-nilmanifolds (see [11,13]). Although we formulated the theory in terms of infra-nilmanifolds in this paper, the reader who is familiar with the the class of infrasolvmanifolds of type (R) will have noticed that all results and proofs directly generalize to this class of manifolds.…”

Formulas for the Reidemeister, Lefschetz and Nielsen coincidence number of maps between infra-nilmanifolds

“…In [11,12], formulas for the Lefschetz (fixed point) number and the Nielsen (fixed point) number of continuous selfmaps on infra-nilmanifolds have been proved. In coincidence theory however, only recently a formula has been proved for the Lefschetz coincidence number and the Nielsen coincidence number of a pair of continuous maps between solvmanifolds of type (R) [13]. For infra-nilmanifolds however, formulas for the Lefschetz coincidence number and the Nielsen coincidence number are still open for study.…”

“…In this article, we address this problem and we prove in Theorem 6.11 explicit and practical formulas for the Reidemeister coincidence number, the Lefschetz coincidence number and the Nielsen coincidence number of a pair of continuous maps between oriented infra-nilmanifolds of equal dimension, generalizing [12] from fixed point theory to coincidence theory and generalizing [13] from nilmanifolds to infra-nilmanifolds. In order to prove these formulas, we use the averaging formulas for the Nielsen coincidence number [14] and for the Lefschetz coincidence number (see [[9], p. 88] and [10]) and we develop an averaging formula for the Reidemeister coincidence number.…”