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Group-theoretic conditions under which closed aspherical manifolds are covered by Euclidean space. (English) Zbl 1055.57032
A well-known problem is to decide which closed aspherical $$m$$-manifolds $$M$$ have universal cover $$\widetilde M$$ homeomorphic to Euclidean space. It is known that if $$m\geq 3$$ (and $$M$$ is irreducible if $$m=3$$), then $$\widetilde M$$ is homeomorphic to Euclidean space if and only if $$\widetilde M$$ is simply connected at infinity, which is a geometric property of $$\pi_1M$$. Hence, it is important to know which finitely presented groups are simply connected at infinity.
Two of the main results of the paper under review are a new geometric proof of a theorem of C. Houghton [J. Lond. Math. Soc., II. Ser. 15, 465–471 (1977; Zbl 0354.20025)] and B. Jackson [Topology 21, 71–81 (1981; Zbl 0472.57001)], namely, if $$1\to H\to G\to Q\to 1$$ is a short exact sequence of finitely presented infinite groups and either $$H$$ or $$Q$$ is one-ended, then $$G$$ is simply connected at infinity, and a generalization of it obtained by relaxing the normality condition on $$H$$ in $$G$$.
The authors also give a geometric proof of the theorem: {If $$M$$ is a closed aspherical $$m$$-manifold (and irreducible if $$m=3$$) and $$\pi_1M$$ contains a non-trivial cyclic normal subgroup, then $$\widetilde{M}$$ is homeomorphic to Euclidean space.}
For $$m\geq 5$$ this is due to R. Lee and F. Raymond [Topology 14, 49–57 (1975; Zbl 0313.57005)] and for $$m=3$$ it follows from results of D. Gabai [Ann. Math. (2) 136, No. 3, 447–510 (1992; Zbl 0785.57004)] and A. Casson and D. Jungreis [Invent. Math. 118, No. 3, 441–456 (1994; Zbl 0840.57005)]. M. Davis [Ann. Math. (2) 117, 293–324 (1983; Zbl 0531.57041)] constructed the first examples of closed aspherical manifolds not covered by Euclidean space and the authors analyze those examples in the light of their results.

##### MSC:
 57N99 Topological manifolds 57N10 Topology of general $$3$$-manifolds (MSC2010) 57N15 Topology of the Euclidean $$n$$-space, $$n$$-manifolds ($$4 \leq n \leq \infty$$) (MSC2010) 57S30 Discontinuous groups of transformations 57M07 Topological methods in group theory 57M10 Covering spaces and low-dimensional topology 20F65 Geometric group theory 20F69 Asymptotic properties of groups
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