Finite covering spaces of 3-manifolds

Following Perelman's solution to the Geometrisation Conjecture, a 'generic' closed 3-manifold is known to admit a hyperbolic structure. However, our understand- ing of closed hyperbolic 3-manifolds is far from complete. In particular, the notorious Virtually Haken Conjecture remains unresolved. This proposes that every closed hyper- bolic 3-manifold has a finite cover that contains a closed embedded orientable �1-injective surface with positive genus. I will give a survey on the progress towards this conjecture and its variants. Along the way, I will address other interesting questions, including: What are the main types of finite covering space of a hyperbolic 3-manifold? How many are there, as a function of the covering degree? What geometric, topological and algebraic properties do they have? I will show how an understanding of various geometric and topological invariants (such as the first eigenvalue of the Laplacian, the rank of mod p homology and the Heegaard genus) can be used to deduce the existence of �1-injective surfaces, and more.

[1]  A. Lubotzky.  Group presentation, $p$-adic analytic groups and lattices in $\mathrm{SL}_2(\mathbf{C})$ , 1983 .

[2]  M. Lackenby.  Surface subgroups of Kleinian groups with torsion , 2008, 0804.1309.

[3]  Growth rates, _{}-homology, and volumes of hyperbolic 3-manifolds , 1992 .

[4]  The asymptoic behaviour of Heegaard genus , 2002, math/0210316.

[5]  Free Groups in Lattices , 2008, 0802.0185.

[6]  J. Rubinstein,et al.  Applications of minimax to minimal surfaces and the topology of 3-manifolds , 1987 .

[7]  N. Dunfield,et al.  Automorphic forms and rational homology 3--spheres , 2005, math/0508271.

[8]  I. Agol,et al.  The Bianchi groups are separable on geometrically finite subgroups , 2001 .

[10]  LERF and the Lubotzky–Sarnak Conjecture , 2008, 0804.1305.

[11]  L. Clozel.  On the cuspidal cohomology of arithmetic subgroups of $\mathrm{SL}(2n)$ and the first Betti number of arithmetic $3$-manifolds , 1987 .

[12]  J. Schwermer,et al.  On liftings and cusp cohomology of arithmetic groups , 1986 .

[13]  B. Weisfeiler.  Strong approximation for Zariski-dense subgroups of semi-simple algebraic groups , 1984 .

[14]  Alexander Lubotzky,et al.  Discrete groups, expanding graphs and invariant measures , 1994, Progress in mathematics.

[15]  W. Lück.  ApproximatingL2-invariants by their finite-dimensional analogues , 1994 .

[16]  Marc Lackenby.  Heegaard splittings, the virtually Haken conjecture and Property (τ) , 2002 .

[17]  J. Bourgain,et al.  Uniform expansion bounds for Cayley graphs of SL2(Fp) , 2008 .

[18]  Igor Nikolaev.  Arithmetic of hyperbolic 3-manifolds , 2002 .

[19]  D. Cooper,et al.  Virtually Haken Dehn-filling , 1999 .

[20]  Daniel T. Wise,et al.  Special Cube Complexes , 2008, The Structure of Groups with a Quasiconvex Hierarchy.

[21]  Marc Lackenby,et al.  Large groups, property (τ) and the homology growth of subgroups , 2009, Mathematical Proceedings of the Cambridge Philosophical Society.

[22]  P. Scott,et al.  The geometries of 3-manifolds , 1983 .

[23]  Detecting large groups , 2007, math/0702571.

[24]  M. Lackenby.  New lower bounds on subgroup growth and homology growth , 2005, math/0512261.

[25]  J. Cheeger.  A lower bound for the smallest eigenvalue of the Laplacian , 1969 .

[26]  A. Lubotzky.  Subgroup growth and congruence subgroups , 1995 .

[27]  P. Buser.  A note on the isoperimetric constant , 1982 .

[28]  D. Solitar,et al.  A note on groups with separable finitely generated subgroups , 1987, Bulletin of the Australian Mathematical Society.

[29]  A. Reid.  A non-Haken hyperbolic $3$-manifold covered by a surface bundle. , 1995 .

[30]  John Stillwell,et al.  Three-Dimensional Manifolds , 1980 .

[31]  EIGENVALUES OF THE LAPLACIAN, THE FIRST BETTI NUMBER AND THE CONGRUENCE SUBGROUP PROBLEM , 1996 .

[32]  M. Lackenby.  Expanders, rank and graphs of groups , 2004, math/0403127.

[33]  Torsion numbers of augmented groups: with applications to knots and links , 2002, math/0202197.

[34]  A. Reid,et al.  Essential closed surfaces in bounded 3-manifolds , 1997 .

[35]  W. Thurston,et al.  The virtual Haken conjecture: Experiments and examples , 2002, math/0209214.

[36]  R. Grigorchuk.  Degrees of Growth of Finitely Generated Groups, and the Theory of Invariant Means , 1985 .

[37]  Pham Do Tuan,et al.  On the estimation of Fourier coefficients. , 1969 .

[38]  Marc Lackenby.  Covering spaces of 3-orbifolds , 2005 .

[39]  N. Boston,et al.  Pro–$p$ groups and towers of rational homology spheres , 2006, 0902.4567.

[40]  T. Venkataramana.  Virtual Betti numbers of compact locally symmetric spaces , 2006 .

[41]  G. Perelman.  The entropy formula for the Ricci flow and its geometric applications , 2002, math/0211159.

[42]  C. McA. Gordon,et al.  Reducing heegaard splittings , 1987 .

[43]  I. Agol.  Criteria for virtual fibering , 2007, 0707.4522.

[44]  Virtually Haken fillings and semi-bundles , 2004, math/0407328.

[45]  Darren D. Long,et al.  Heegaard genus and property τ for hyperbolic 3‐manifolds , 2008 .

[46]  D. Sullivan.  Related aspects of positivity in Riemannian geometry , 1987 .

[47]  G. Perelman.  Finite extinction time for the solutions to the Ricci flow on certain three-manifolds , 2003, math/0307245.

[48]  W. Thurston.  Three dimensional manifolds, Kleinian groups and hyperbolic geometry , 1982 .

[49]  V. Marković,et al.  Immersing almost geodesic surfaces in a closed hyperbolic three manifold , 2009, 0910.5501.

[50]  C. Leininger.  Surgeries on one component of the Whitehead link are virtually fibered , 2002 .

[51]  Robert F. Riley.  Growth of order of homology of cyclic branched covers of knots , 1990 .

[52]  I. Agol,et al.  Virtually fibered Montesinos links , 2008 .

[53]  G. Perelman.  Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[54]  T. Rassias.  EIGENVALUES OF THE LAPLACIAN , 1991 .

[55]  H. Short,et al.  Cyclic Branched Coverings of Knots and Homology Spheres , 1991 .

[56]  Martin Scharlemann,et al.  Chapter 18 – Heegaard Splittings of Compact 3-Manifolds , 2000, math/0007144.

[57]  A. Reid,et al.  On the virtual Betti numbers of arithmetic hyperbolic 3-manifolds , 2007 .

[58]  D. Long.  Immersions and Embeddings of Totally Geodesic Surfaces , 1987 .