A REMARK ON 4-MANIFOLDS WITH ZERO ENTROPY METRICS

J. A. HILLMAN

Abstract. We sharpen a recent result of Paternain and Petean by showing that a 4-manifold which admits a Riemannian metric with zero topological entropy either has a finite covering space homeomorphic to S^{3} × S^{1} or S^{2} × T or is aspherical.

Paternain and Petean showed that if a 4-manifold admits a Riemannian metric with zero topological entropy then its universal covering space has the rational homotopy type of S^{2} or S^{3} or of a point [7]. They show also that if π_{1}(M) has polynomial growth then M has a finite covering space homeomorphic to S^{3} × S^{1}, S^{2} × T or a nilmanifold. We may sharpen this result as follows.

Theorem. Let M be a 4-manifold that admits a Riemannian metric with zero topological entropy. Then either M has a finite covering space homeomorphic to S^{3} × S^{1} or S^{2} × T or it is aspherical. In the latter case, if π = π_{1}(M) has a subgroup σ of finite index with β_{1}(σ) ≥ 2 then it is virtually nilpotent and M is homeomorphic to an infranilmanifold.

Proof. We may clearly assume that M is orientable in order to prove the first assertion. The zero entropy hypothesis implies that π has subexponential growth [3]. Moreover χ(M) = 0, by Theorem 4.1 of [7], and so π maps onto Z (cf. Lemma 3.14 of [5]). Hence π is an HNN extension HNN(B; ϕ : I → J) with finitely generated base B [1]. Such an HNN extension has exponential growth unless I = J = B (see page 196 of [2]). Hence π ≅ B ⋊ Z. If B is finite or has two ends then π is virtually Z or Z^{2}, and the result follows from Theorems 10.10 and 11.1 of [5]. Otherwise B has one end (since it has subexponential growth), and so M is aspherical and B is a P D_{3}^{+}-group, by Corollary 6.3 of [6].

If π has a subgroup σ of finite index with β_{1}(σ) ≥ 2 then B_{1} = B ∩ σ has finite index in B and maps onto Z. Since B_{1} is F P_{2} it is an HNN extension with finitely generated base C [1], and since it has subexponential growth it is a semidirect product B_{1} ≅ C ⋊ Z [2]. Hence C must be a P D_{2}^{+}-group, by Corollary 6.1 of [6]. Moreover C ≅ Z^{2}, since it has subexponential growth. Therefore π is virtually solvable, and hence virtually nilpotent [4]. The final assertion follows from Theorem 8.1 of [5]. □

Must π be virtually nilpotent in all cases? This would be so if every P D_{3}-group has a subgroup of finite index with infinite abelianization.

1991 Mathematics Subject Classification. 53D25,37B40.

Key words and phrases. entropy, 4-manifold, infranilmanifold, subexponential growth.

References

[1] Bieri, R. and Strebel, R. Almost finitely presentable soluble groups, Comment. Math. Helv. 53 (1978), 258-278.

[2] de la Harpe, P. and Bucher, M. Free products with amalgamation and HNN extensions which are of uniformly exponential growth, Mat. Zametki 67 (2000), 811-815 (Russian). English translation: Math. Notes 67 (2000), 686-689.

[3] Dinaburg, E.I. On the relations among various entropy characteristics of dynamical systems, Math. USSR Izv. 5 (1971), 337-378.

[4] Gromov, M. Groups of polynomial growth and expanding maps, IHES Publ. Math.53 (1981), 53-73.

[5] Hillman, J.A. Four-Manifolds, Geometries and Knots, Geometry and Topology Monographs Vol. 5 (2002).

[6] Hillman, J.A. and Kochloukova, D. Finiteness conditions and P D_{r}-group covers of P D_{n}- complexes, preprint, University of Sydney/UNICAMP-IMECC (2005).

[7] Paternain, G.P. and Petean, J. Zero entropy and bounded topology, preprint, Cambridge University/CIMAT (2005).

School of Mathematics and Statistics,, University of Sydney, NSW 2006, AUSTRALIA

E-mail address: jonh@maths.usyd.edu.au