Four-dimensional graph-manifolds with fundamental groups quasiisometric to fundamental groups of orthogonal graph-manifolds

Smirnov, A.

doi:10.1090/spmj/1526

Four-dimensional graph-manifolds with fundamental groups quasiisometric to fundamental groups of orthogonal graph-manifolds
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by A. Smirnov
Translated by: The author

St. Petersburg Math. J. 29 (2018), 1031-1043

DOI: https://doi.org/10.1090/spmj/1526

Published electronically: September 4, 2018

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Abstract:

A topological invariant called the type of a graph-manifold, which takes natural values, is introduced. For a 4-dimensional graph-manifold whose type does not exceed two it is proved that its universal cover is bi-Lipschitz equivalent to a universal cover of an orthogonal graph-manifold (for arbitrary Riemannian metrics on graph-manifolds).

References

Friedhelm Waldhausen, Eine Klasse von $3$-dimensionalen Mannigfaltigkeiten. I, II, Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117 (German). MR 235576, DOI 10.1007/BF01402956
Friedhelm Waldhausen, Eine Klasse von $3$-dimensionalen Mannigfaltigkeiten. I, II, Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117 (German). MR 235576, DOI 10.1007/BF01402956
S. V. Buyalo and V. L. Kobel′skiĭ, Generalized graph-manifolds of nonpositive curvature, Algebra i Analiz 11 (1999), no. 2, 64–87 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 11 (2000), no. 2, 251–268. MR 1702579
Sergei Buyalo and Viktor Schroeder, Elements of asymptotic geometry, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. MR 2327160, DOI 10.4171/036
M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
David Hume and Alessandro Sisto, Embedding universal covers of graph manifolds in products of trees, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3337–3340. MR 3080156, DOI 10.1090/S0002-9939-2013-11669-2
M. Kapovich and B. Leeb, $3$-manifold groups and nonpositive curvature, Geom. Funct. Anal. 8 (1998), no. 5, 841–852. MR 1650098, DOI 10.1007/s000390050076
Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
A. V. Smirnov, Quasi-isometric embedding of the fundamental group of an orthogonal graph manifold into a product of trees, Algebra i Analiz 24 (2012), no. 5, 165–180 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 24 (2013), no. 5, 811–821. MR 3087825, DOI 10.1090/S1061-0022-2013-01267-2