Heegaard splittings, the virtually Haken conjecture and Property (τ)

By Marc Lackenby. 2002

94 citations recorded by the source corpus.

The behaviour of finite covers of 3-manifolds is a very important, but poorly understood, topic. There are three, increasingly strong, conjectures in the field that have remained open for over twenty years – the virtually Haken conjecture, the positive virtual b1 conjecture and the virtually fibred conjecture. Any of these would have profound ramifications for 3-manifold theory. In this paper, we explore the interaction of these conjectures with the following seemingly unrelated areas: eigenvalues of the Laplacian, and Heegaard splittings.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ed4b619928c4de819efd4feeefd38b901cc85df2

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Some topics in the dynamics of group actions on rooted trees (2011, methodology): “The concept of rank gradient was introduced by Lackenby [121] in connection with the research on three-dimensional topology [121, 122].”
KLEINIAN GROUPS IN HIGHER DIMENSIONS (2007, methodology): “ere ρ : Γ →Isom(Hn+1) is the natural embedding. On the other hand, the proofs of special cases of Conjecture 11.11 for hyperbolic 3-manifolds which use the methods of 3-dimensional topology (see e.g. [141]), usually provide no information about rigidity of Γ in Isom(H4). 49 Conjecture 11.12. Suppose”
A SURVEY OF MEASURED GROUP THEORY (2009, background): “Surprisingly, this turns out to be related to a problem in the theory of compact hyperbolic 3-manifolds concerning rank versus Heegard genus [89] – see [1] for the connection and further discussions.”
Expansion in perfect groups (2011, result): “velopments. Besides sieving, results similar to the above theorem are useful in studying covers of hyperbolic 3-manifolds, see the paper of Long, Lubotzky and Reid [46] following the work of Lackenby [43]. The question about the expanding properties of mod qquotients was ﬁrst studied only for “thick”
Expanders, rank and graphs of groups (2004, background): “The purpose of Theorem 1.7 in [ 4 ] is that 360 M. LACKENBY Isr.”
The virtual Haken conjecture (2012, background): “…covering space has a finite hierarchy obtained by labeling the walls with finitely many numbers (which we think of as colors), so that walls with the same color do not intersect, and cutting successively along the walls ordered by their labels to get an infinite collection of “cubical polyhedra”.”
Heegaard Splittings and Pseudo-Anosov Maps (2009, background): “[ Lac , Sou] If F is a minimal surface of genus g in a complete Riemannian manifold M with curvature M 1 2 and injectivity radius inj(M) then one has”
Rank gradient, cost of groups and the rank versus Heegaard genus problem (2007, background): “The boundary of M is the negative boundary @B1− ∪@B2−. See [ 17 ], Section 3 for more details.”
Nilprogressions and groups with moderate growth (2015, unknown): “As a consequence of Theorem 1.3, we can remove the finite presentability assumption and strengthen Lackenby’s result as follows.”