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http://dx.doi.org/10.4134/BKMS.b190601

NIELSEN SPECTRUM OF MAPS ON INFRA-SOLVMANIFOLDS MODELED ON Sol04  

Lee, Jong Bum (Department of Mathematics Sogang University)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.4, 2020 , pp. 909-919 More about this Journal
Abstract
The 4-dimensional solvable Lie group Sol04 does not admit a lattice. The purpose of this paper is two-fold. We study poly-crystallographic groups of Sol04, and then we study Nielsen fixed point theory on the spaces modeled on Sol04.
Keywords
Infra-solvmanifold; Lefschetz number; mapping torus; Nielsen number; poly-crystallographic group; Reidemeister number; $Sol_0{^4}$;
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Times Cited By KSCI : 5  (Citation Analysis)
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1 K. Y. Ha and J. B. Lee, Crystallographic groups of Sol, Math. Nachr. 286 (2013), no. 16, 1614-1667. https://doi.org/10.1002/mana.201200304   DOI
2 K. Y. Ha and J. B. Lee, The $R_{\infty}$ property for crystallographic groups of Sol, Topology Appl. 181 (2015), 112-133. https://doi.org/10.1016/j.topol.2014.12.005   DOI
3 K. Y. Ha and J. B. Lee, The Nielsen type numbers for maps on a 3-dimensional at Riemannian manifold, Topol. Methods Nonlinear Anal. 45 (2015), no. 2, 327-362. https://doi.org/10.12775/TMNA.2015.017   DOI
4 K. Y. Ha and J. B. Lee, Averaging formula for Nielsen numbers of maps on infra-solvmanifolds of type (R)-corrigendum, Nagoya Math. J. 221 (2016), 207-212.   DOI
5 K. Y. Ha, J. B. Lee, and P. Penninckx, Formulas for the Reidemeister, Lefschetz and Nielsen coincidence number of maps between infra-nilmanifolds, Fixed Point Theory Appl. 2012 (2012), 39, 23 pp. https://doi.org/10.1186/1687-1812-2012-39   DOI
6 M. Inoue, New surfaces with no meromorphic functions, in Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, 423-426, Canad. Math. Congress, Montreal, QC, 1975.
7 B. J. Jiang, Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics, 14, American Mathematical Society, Providence, RI, 1983.
8 J. H. Jo and J. B. Lee, Nielsen type numbers and homotopy minimal periods for maps on solvmanifolds with Sol41-geometry, Fixed Point Theory Appl. 2015 (2015), 175, 15 pp. https://doi.org/10.1186/s13663-015-0427-x   DOI
9 J. H. Jo and J. B. Lee, Nielsen fixed point theory on infra-solvmanifolds of Sol, Topol. Methods Non- linear Anal. 49 (2017), no. 1, 325-350. https://doi.org/10.12775/tmna.2016.080
10 S. W. Kim, J. B. Lee, and K. B. Lee, Averaging formula for Nielsen numbers, Nagoya Math. J. 178 (2005), 37-53. https://doi.org/10.1017/S0027763000009107   DOI
11 H. J. Kim, J. B. Lee, and W. S. Yoo, Computation of the Nielsen type numbers for maps on the Klein bottle, J. Korean Math. Soc. 45 (2008), no. 5, 1483-1503. https://doi.org/10.4134/JKMS.2008.45.5.1483   DOI
12 J. B. Lee and K. B. Lee, Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds, J. Geom. Phys. 56 (2006), no. 10, 2011-2023. https://doi.org/10.1016/j.geomphys.2005.11.003   DOI
13 J. B. Lee and K. B. Lee, Averaging formula for Nielsen numbers of maps on infra-solvmanifolds of type (R), Nagoya Math. J. 196 (2009), 117-134. https://doi.org/10.1017/S0027763000009818   DOI
14 J. B. Lee and X. Zhao, Density of the homotopy minimal periods of maps on infra-solvmanifolds of type (R), J. Korean Math. Soc. 55 (2018), no. 2, 293-311. https://doi.org/10.4134/JKMS.j170189   DOI
15 J. B. Lee, K. B. Lee, J. Shin, and S. Yi, Unimodular groups of type $R^3{\times}R$, J. Korean Math. Soc. 44 (2007), no. 5, 1121-1137. https://doi.org/10.4134/JKMS.2007.44.5.1121   DOI
16 J. B. Lee and W. S. Yoo, Nielsen theory on nilmanifolds of the standard filiform Lie group, Trends in Algebraic Topology and Related Topics, Trends Math., pp. 177-195, Springer Basel AG, Basel, 2019.
17 J. B. Lee and X. Zhao, Homotopy minimal periods for expanding maps on infra-nilmanifolds, J. Math. Soc. Japan 59 (2007), no. 1, 179-184. http://projecteuclid.org/euclid.jmsj/1180135506   DOI
18 K. B. Lee and F. Raymond, Seifert Fiberings, Mathematical Surveys and Monographs, 166, American Mathematical Society, Providence, RI, 2010. https://doi.org/10.1090/surv/166
19 K. B. Lee and S. Thuong, Infra-solvmanifolds of $Sol^4_1$, J. Korean Math. Soc. 52 (2015), no. 6, 1209-1251. https://doi.org/10.4134/JKMS.2015.52.6.1209   DOI
20 S. V. Thuong, Classification of closed manifolds with $Sol^4_1$-geometry, Geom. Dedicata 199 (2019), 373-397. https://doi.org/10.1007/s10711-018-0354-1   DOI
21 A. Fel'shtyn and J. B. Lee, The Nielsen numbers of iterations of maps on infra-solvmanifolds of type (R) and periodic orbits, J. Fixed Point Theory Appl. 20 (2018), no. 2, Art. 62, 31 pp. https://doi.org/10.1007/s11784-018-0541-6
22 B. Wilking, Rigidity of group actions on solvable Lie groups, Math. Ann. 317 (2000), no. 2, 195-237. https://doi.org/10.1007/s002089900091   DOI
23 C. T. C. Wall, Geometric structures on compact complex analytic surfaces, Topology 25 (1986), no. 2, 119-153. https://doi.org/10.1016/0040-9383(86)90035-2   DOI
24 Y. Choi, J. B. Lee, and K. B. Lee, Nielsen theory on infra-nilmanifolds modeled on the group of uni-triangular matrices, Fixed Point Theory, 20 (2019), 438-506.
25 A. Fel'shtyn and J. B. Lee, The Nielsen and Reidemeister numbers of maps on infra-solvmanifolds of type (R), Topology Appl. 181 (2015), 62-103. https://doi.org/10.1016/j.topol.2014.12.003   DOI
26 A. Fel'shtyn and J. B. Lee, The Nielsen and Reidemeister theories of iterations on infra-solvmanifolds of type (R) and poly-Bieberbach groups, in Dynamics and numbers, 77-103, Contemp. Math., 669, Amer. Math. Soc., Providence, RI, 2016. https://doi.org/10.1090/conm/669/13424