Is every fundamental group of a mapping torus of a surface homeomorphism a CAT(0) group?

Question

TLDR; Is every fundamental group of a mapping torus of a surface homeomorphism a CAT(0) group?

Let $S$ be a compact surface of negative Euler characteristic and let $f :S\to S$ be a homeomorphism. Write $M_f$ to denote the mapping torus of $f$, $$M_f = \frac{S \times [0,1]}{(0,f(x)) \sim (1, x)}.$$

Thurston shows that:

1. If $f$ is pseudo-Anosov then $M_f$ admits a complete hyperbolic structure of finite volume.
2. If $f$ is finite order then $M_f$ admits a complete $\mathbb{H}^2 \times \mathbb{R}$ structure.
3. If $f$ is reducible then $M_f$ admits an embedded incompressible torus, not parallel to the boundary.

If $S$ has empty boundary, then in cases 1 and 2 we can conclude that $M_f$ acts properly and cocompactly on $\mathbb{H}^{3}$ and $\mathbb{H}^2 \times \mathbb{R}$, respectively, and thus that $\pi_1(M_f)$ is CAT(0).

My question: What can be said in the case when $S$ (and thus $M_f$) has non-empty boundary? What can be said in case 3?

Answer 1

In Cases 1 and 2 with nonempty boundary, the complete metric on $M_f$ can be truncated to give a metric on a compact space, by "cutting off horoballs". In Case 1 you cut off a $\pi_1 M_f$-invariant, pairwise disjoint family of open, 3-dimensional hyperbolic horoballs in $\mathbb H^3$, leaving $\mathbb R^2$ boundaries, and then mod out by the group, leaving torus boundaries. In Case 2 you cut off a $\pi_1(S)$-invariant, pairwise disjoint family of open 2-dimensional hyperbolic horodiscs in the $\mathbb H^2$ factor, leaving $\mathbb R$ boundaries in that factor, and then cross with the $\mathbb R$ factor, leaving $\mathbb R^2$ boundaries; and then again you mod out by the group leaving torus boundaries. In either case, the restricted path metric is a CAT(0) metric (this is probably explained in the book of Bridson and Haefliger).

In Case 3, there's some literature you can consult. I can't find a complete description, and the totality of what I can find canot be squeezed into an answer, but there's several results in this paper of Bernard Leeb as a start. For example, in the case of empty boundary, Theorem 3.3 of that paper, combined with Thurston's geometrization theorem, implies that if at least one component of the Thurston decomposition of $f$ is pseudo-Anosov then $M_f$ admits a CAT(0) geometry.

Case 3 get more complicated in the subcase that every component of the Thurston decomposition of $f$ is finite order, and so every component of the torus decomposiiton of $M_f$ is Seifert fibered. Dropping for a moment the hypothesis that $M$ is a mapping torus, Leeb's paper describes an examples of a closed, irreducible manifold $M$ whose torus decomposition has only Seifert fibered pieces, and where $M$ itself has no locally CAT(0) metric, and yet $\pi_1(M)$ is a CAT(0) group. But I do not know whether such examples exist where $M$ is a mapping torus, and I do not know whether any mapping torus examples exist where the group is not CAT(0).

Answer 2

Yes, there are examples of mapping tori which are not CAT(0). The simplest one is when $f$ is a single Dehn twist.

See Theorem 3.7 in

Kapovich, Michael; Leeb, Bernhard, Actions of discrete groups on nonpositively curved spaces, Math. Ann. 306, No. 2, 341-352 (1996). ZBL0856.20024.

You would also need Theorem 2.4 in their paper.

The ultimate result is contained in

Buyalo, S.; Svetlov, P., Topological and geometric properties of graph-manifolds, St. Petersbg. Math. J. 16, No. 2, 297-340 (2005); translation from Algebra Anal. 16, No. 2, 3-68 (2004). ZBL1082.57015.

In this paper you will find a combinatorial criterion for the mapping torus of a completely reducible homeomorphism of a surface to admit a metric of nonpositive curvature. (I find their condition to be too complicated to be discussed here.) See also references there to the earlier papers by Buyalo and Kobelsky on this subject. One of the more surprising results they prove in the paper (it is actually due to Svetov) is that the fundamental group of a closed graph-manifold is CAT(0) if and only if the manifold virtually fibers over the circle.