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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Ku Yong Ha, Jong Bum Lee and Pieter Penninckx
Journal: Proc. Amer. Math. Soc. 139 (2011), 2239-2248
MSC (2010): Primary 55M20, 54H25, 57S30
Published electronically: November 5, 2010
MathSciNet review: 2775401
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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:

$\displaystyle 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.


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Additional Information

Ku Yong Ha
Affiliation: Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
Email: kyha@sogang.ac.kr

Jong Bum Lee
Affiliation: Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
Email: jlee@sogang.ac.kr

Pieter Penninckx
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Campus Kortrijk, B-8500 Kortrijk, Belgium
Email: pieter.penninckx@kuleuven-kortrijk.be

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10721-9
PII: S 0002-9939(2010)10721-9
Keywords: Nielsen coincidence number, solvmanifold
Received by editor(s): August 11, 2009
Received by editor(s) in revised form: May 31, 2010
Published electronically: November 5, 2010
Additional Notes: The second-named author is supported partially by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R01-2007-000-10097-0) and by the Sogang University Research Grant of 2010.
The third-named author is supported by a Ph.D. fellowship of the Research Foundation-Flanders (FWO)
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.