Low Dimensional Topology

June 27, 2008

Commensurability

Filed under: Uncategorized — Jesse Johnson @ 11:16 am

Here’s an interesting construction that I recently encountered: It’s possible to find a knot in the 3-sphere whose complement has a finite cover that is also a 3-sphere knot complement. Let K be a knot with a Dehn surgery producing a lens space (for example a Berge knot). The lens space is finitely covered by the 3-sphere and the image of K lifted to the 3-sphere is a new knot K’ . The complement of K’ finitely covers the complement of K. It turns out this is the only way to build a knot complement covered by a knot complement, which is proved in [1]. It also shows up in Reid and Walsh’s paper on commensurability classes of 2-bridge knots [2]. (For the record, it was grad. student Neil Hoffman of UT Austin who told me about this construction.)

Two compact 3-manifolds are called commensurable if one has a finite cover that is homeomorphic to a finite cover of the other. Two groups are called commensurable if one has a finite index subgroup that is isomorphic to a finite index subgroup of the other. I don’t know which definition came first, but thanks to some basic algebraic topology, the definitions are more or less equivalent: A cover of a topological space is uniquely determined by (the conjugacy class of) a subgroup of the fundamental group. Thus if two 3-manifolds are commensurable then their fundamental groups are commensurable. Conversely, a closed (or cusped) hyperbolic 3-manifold is determined by its fundamental group. Thus two hyperbolic 3-manifolds have commensurable fundamental groups if and only if they’re commensurable. Note that commensurable groups are quasi-isometric, so these ideas are related to coarse geometry as well.

Walsh and Reid prove their result by showing that in the commensurability class for a knot complement, there is a unique minimal element (i.e. it is covered by every 3-manifold in the class) that is the quotient of the hyperbolic plane by the normalizer in Isom(H^3) of the group of isomitries that produce the knot complement. (The knot complement is a regular cover of this manifold, so they need to show that the knot complement doesn’t have any hidden symmetries, i.e. doesn’t irregularly cover any smaller 3-manifold.) They then show that this minimal element of the commensurability class covers exactly one knot complement.

[1] F. Gonzalez-Acuna and W. C. Whitten, Imbeddings of three-manifold groups,
Mem. Amer. Math. Soc. 474 (1992).