A Folded Mapping Torus

A Familiar Story

Here's one (perhaps familiar) way to construct a 3-manifold M:

Begin with your favorite surface S.
Find a homeomorphism F on S.
Let M be S x [0,1] with each (s ,1) identified with (F(s),0).

M is called the mapping torus of F. When S is a hyperbolic surface of finite type, M is hyperbolic if and only if the homeomorphism F is pseudo-Anosov (roughly, pseudo-Ansov means F is dynamically rich. Lots of mixing happens under iterations of F).

From Surfaces to Graphs

Homeomorphisms on surfaces (usually up to isotopy) present a rich area of study, including the study of their mapping tori. The group of homeomorphisms on a given surface S up to isotopy is called the mapping class group of S, denoted here MCG(S).

If S is a surface with genus g and n boundary components, there is a deformation retraction of S to a graph of rank 2g+n-1. So in some sense, finite graphs are a degeneration of finite type surfaces. However, the group of homeomorphisms of a finite graph G up to isotopy is much less interesting (from the perspective of a geometric group theorist) than MCG(S). A more appropriate analogy of MCG(S) for a finite graph G is the group of homotopy equivalences on G up to isotopy.

Suppose F is a homotopy equivalence on a rank r graph G. Since F may not fix any base point in G, F induces an outer automorphism on the fundamental group of G. As pi1(G) is isomorphic to the free group of rank r, F corresponds to an element of Out(Fr) := Aut(Fr) / Inn(Fr). It turns out this outer automorphism encodes all the important information we would like to retain about F: that is, Out(Fr) plays the role for graphs that MCG(S) plays for surfaces.

When S is a surface with boundary, every element in MCG(S) corresponds to an element in Out(Fr). However, for rank > 2, Out(Fr) has many many elements beyond those coming from some mapping class group element. Informally, the intuition behind this is that continuity on a graph is a much less restrictive than continuity on a surface.

Mapping Tori of Outer Automorphisms

In line with the familiar story above, we can build a mapping torus M of a given homotopy equivalence graph map F on a graph G of rank r:

Consider G x [0,1].
Identify (g,1) with (F(g),0).

The fundamental group of M is a free-by-cyclic group. One disadvantage of defining M in this way is the interesting geometry of the mapping torus is hidden in the identification. To uncover this geometry, we can decompose F into pieces and incorporate each piece into the "visible" part of the mapping torus, so the identification is just the identity.

Fold Decomposition

Every surjective graph map F can be written as a composition of "folds" and a graph isomorphism.

A fold is an identification of one edge in the graph (or one part of an edge) with another adjacent edge (or part of an adjacent edge).

Each fold can be interpreted as a smooth deformation of one graph into the next.

Folded Mapping Torus

Suppose F is the composition of:

a fold F1 from G to G',
a fold F2 from G' to G'', and
an isomorphism H from G'' to G.

For t in [0,1], let Gt be the smooth deformation of G into G' and for t in [1,2] let Gt be the smooth deformation of G' into G''.

The folded mapping torus of F is { (g,t) | g in Gt and t in [0,2] } with each (g,2) identified with (H(g),0). Further, we can draw the isomorphism H from G'' to G' beneath the final fold, so that visually the identification is given by the identity.

An Example

To the right is the fold decomposition of a graph map consisting of one partial fold and a graph isomorphism. Below is the corresponding folded mapping torus.

The 1-Skeleton

Folded mapping torus

Entire folded mapping torus

When is the Mapping Torus Hyperbolic?

While in this case the mapping torus is not a manifold, we can still ask whether the mapping torus of a given homotopy equivalence F on a graph is hyperbolic. This is the same as asking when the corresponding free by cyclic group is a hyperbolic group. The answer turns out to be similar to that in the familiar story: The mapping torus is hyperbolic if and only if the outer automorphism determined by F is atoriodal, meaning no power preserves a conjugacy class of a nontrivial element of the free group.

From this perspective, the atoriodal elements of Out(Fr) play the role of the pseudo-Anosov elements of MCG(S). However this analogy breaks down when comparing Out(Fr) and MCG(S) from other perspectives. For example, Out(Fr) acts on rank r Culler-Vogtmann Outer Space similarly to how MCG(S) acts on the Teichmuller space of S. The pseudo-Anosov elements of MCG(S) are exactly those which act loxodromically on Teichmuller space. However, atoriodal elements of Out(Fr) may or may not act loxodromically on outer space, and there are toriodal elements which do act loxodromically.

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