Expanding maps on infranilmanifolds of homogeneous type
Authors:
Karel Dekimpe and Kyung Bai Lee
Journal:
Trans. Amer. Math. Soc. 355 (2003), 10671077
MSC (2000):
Primary 37D20; Secondary 17B30, 17B70
Published electronically:
October 24, 2002
MathSciNet review:
1938746
Fulltext PDF Free Access
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Abstract: In this paper we investigate expanding maps on infranilmanifolds. Such manifolds are obtained as a quotient , where is a connected and simply connected nilpotent Lie group and is a torsionfree uniform discrete subgroup of , with a compact subgroup of . We show that if the Lie algebra of is homogeneous (i.e., graded and generated by elements of degree 1), then the corresponding infranilmanifolds admit an expanding map. This is a generalization of the result of H. Lee and K. B. Lee, who treated the 2step nilpotent case.
 1.
Dekimpe, K.
AlmostBieberbach Groups: Affine and Polynomial Structures, volume 1639 of Lecture Notes in Math., SpringerVerlag, 1996. MR 2000b:20066
 2.
Dekimpe, K. and Malfait, W.
Affine structures on a class of virtually nilpotent groups. Topol. and its Applications, 1996, 73 (2), pp. 97119. MR 97j:57060
 3.
Gromov, M.
Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math., 1981, 53, pp. 5373. MR 83b:53041
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Hall, P.
The Edmonton Notes on Nilpotent Groups. Queen Mary College Math. Notes, London, 1969. MR 44:316
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Johnson, R. W.
Homogeneous Lie Algebras and Expanding Automorphisms. Proc. Amer. Math. Soc., 1975, 48 (2), pp. 292296. MR 51:10417
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Kargapolov, M. and Merzljakov, J.
Fundamentals of the Theory of Groups, volume 62 of Grad. Texts in Math. SpringerVerlag, 1979. MR 80k:20002
 7.
Lee, H. and Lee, K. B.
Expanding maps on 2step infranilmanifolds. Topology Appl., 2002, 117 (1), pp. 4558. MR 2002i:57038
 8.
Lee, K. B. and Raymond, F.
Rigidity of almost crystallographic groups. Contemporary Math. Amer. Math. Soc., 1985, 44, pp. 7378. MR 87d:57026
 9.
Leger, G.
Derivations of Lie Algebras III. Duke Math. J., 1963, 30 (4), pp. 637645. MR 28:3064
 10.
Mal'cev, A. I.
On a class of homogeneous spaces. Translations Amer. Math. Soc., 1951, 39, pp. 133. MR 12:589e
 11.
Segal, D.
Polycyclic Groups. Cambridge Tracts in Mathematics, No. 82, Cambridge University Press, 1983. MR 85h:20003
 1.
 Dekimpe, K.
AlmostBieberbach Groups: Affine and Polynomial Structures, volume 1639 of Lecture Notes in Math., SpringerVerlag, 1996. MR 2000b:20066
 2.
 Dekimpe, K. and Malfait, W.
Affine structures on a class of virtually nilpotent groups. Topol. and its Applications, 1996, 73 (2), pp. 97119. MR 97j:57060
 3.
 Gromov, M.
Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math., 1981, 53, pp. 5373. MR 83b:53041
 4.
 Hall, P.
The Edmonton Notes on Nilpotent Groups. Queen Mary College Math. Notes, London, 1969. MR 44:316
 5.
 Johnson, R. W.
Homogeneous Lie Algebras and Expanding Automorphisms. Proc. Amer. Math. Soc., 1975, 48 (2), pp. 292296. MR 51:10417
 6.
 Kargapolov, M. and Merzljakov, J.
Fundamentals of the Theory of Groups, volume 62 of Grad. Texts in Math. SpringerVerlag, 1979. MR 80k:20002
 7.
 Lee, H. and Lee, K. B.
Expanding maps on 2step infranilmanifolds. Topology Appl., 2002, 117 (1), pp. 4558. MR 2002i:57038
 8.
 Lee, K. B. and Raymond, F.
Rigidity of almost crystallographic groups. Contemporary Math. Amer. Math. Soc., 1985, 44, pp. 7378. MR 87d:57026
 9.
 Leger, G.
Derivations of Lie Algebras III. Duke Math. J., 1963, 30 (4), pp. 637645. MR 28:3064
 10.
 Mal'cev, A. I.
On a class of homogeneous spaces. Translations Amer. Math. Soc., 1951, 39, pp. 133. MR 12:589e
 11.
 Segal, D.
Polycyclic Groups. Cambridge Tracts in Mathematics, No. 82, Cambridge University Press, 1983. MR 85h:20003
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Additional Information
Karel Dekimpe
Affiliation:
Katholieke Universiteit Leuven Campus Kortrijk, Universitaire Campus, B8500 Kortrijk, Belgium
Email:
Karel.Dekimpe@kulak.ac.be
Kyung Bai Lee
Affiliation:
University of Oklahoma, Norman, Oklahoma 73019
Email:
kblee@math.ou.edu
DOI:
http://dx.doi.org/10.1090/S0002994702030842
PII:
S 00029947(02)030842
Keywords:
Infranilmanifold,
expanding map,
homogeneous Lie group
Received by editor(s):
December 11, 2000
Received by editor(s) in revised form:
March 15, 2002
Published electronically:
October 24, 2002
Article copyright:
© Copyright 2002
American Mathematical Society
