Expanding maps on infra-nilmanifolds of homogeneous type

Dekimpe, Karel; Lee, Kyung

doi:10.1090/S0002-9947-02-03084-2

Expanding maps on infra-nilmanifolds of homogeneous type
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Trans. Amer. Math. Soc. 355 (2003), 1067-1077 Request permission

Abstract:

In this paper we investigate expanding maps on infra-nilmanifolds. Such manifolds are obtained as a quotient $E\backslash L$, where $L$ is a connected and simply connected nilpotent Lie group and $E$ is a torsion-free uniform discrete subgroup of $L {\mathbb o} C$, with $C$ a compact subgroup of $\operatorname {Aut}(L)$. We show that if the Lie algebra of $L$ is homogeneous (i.e., graded and generated by elements of degree 1), then the corresponding infra-nilmanifolds admit an expanding map. This is a generalization of the result of H. Lee and K. B. Lee, who treated the 2-step nilpotent case.

References

Karel Dekimpe, Almost-Bieberbach groups: affine and polynomial structures, Lecture Notes in Mathematics, vol. 1639, Springer-Verlag, Berlin, 1996. MR 1482520, DOI 10.1007/BFb0094472
Karel Dekimpe and Wim Malfait, Affine structures on a class of virtually nilpotent groups, Topology Appl. 73 (1996), no. 2, 97–119. MR 1416754, DOI 10.1016/0166-8641(96)00069-7
Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
Philip Hall, The Edmonton notes on nilpotent groups, Queen Mary College Mathematics Notes, Queen Mary College, Mathematics Department, London, 1969. MR 0283083
R. Warren Johnson, Homogeneous Lie algebras and expanding automorphisms, Proc. Amer. Math. Soc. 48 (1975), 292–296. MR 374217, DOI 10.1090/S0002-9939-1975-0374217-4
M. I. Kargapolov and Ju. I. Merzljakov, Fundamentals of the theory of groups, Graduate Texts in Mathematics, vol. 62, Springer-Verlag, New York-Berlin, 1979. Translated from the second Russian edition by Robert G. Burns. MR 551207
Hyunkoo Lee and Kyung Bai Lee, Expanding maps on 2-step infra-nilmanifolds, Topology Appl. 117 (2002), no. 1, 45–58. MR 1874083, DOI 10.1016/S0166-8641(00)00104-8
Kyung Bai Lee and Frank Raymond, Rigidity of almost crystallographic groups, Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982) Contemp. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985, pp. 73–78. MR 813102, DOI 10.1090/conm/044/813102
G. Leger, Derivations of Lie algebras. III, Duke Math. J. 30 (1963), 637–645. MR 159848
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
Daniel Segal, Polycyclic groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983. MR 713786, DOI 10.1017/CBO9780511565953