Discrete fundamental groups of warped cones and expanders

Abstract

In this paper we compute the discrete fundamental groups of warped cones. As an immediate consequence, this allows us to show that there exist coarsely simply-connected expanders and superexpanders. This also provides a strong coarse invariant of warped cones and implies that many warped cones cannot be coarsely equivalent to any box space.

Appendix A: Discrete fundamental group for (unified) warped cones

In this appendix we wish to show that much of the work here developed for warped systems can be adapted to warped cones as they were originally defined in [27]. We begin by properly recalling the definition: given a warped system $\mathrm{WSys}(S\curvearrowright X)$, the action of $F_S$ trivially extends to a level-preserving action on the direct product $X\times [1,\infty )$. The space $X\times [1,\infty )$ can be equipped with a ‘conical’ metric $d_\mathrm{cone}$ in various (roughly equivalent) ways. For example, if (X, g) is a Riemannian manifold it is natural to define $d_\mathrm{cone}$ by $t\cdot g+dt^2$ (this is Roe’s original definition). For more general spaces one can do as in [28] or use the 0-cone metric as in [3, Chapter I.5]. The (unified) warped cone$\mathcal {O}_S (X)$ is the space where is the warped metric obtained warping $d_\mathrm{cone}$ with the $F_S$-action.

As in the introduction, we denote by $\mathcal {O}_S^t(X)$ the subset $X\times \{t\}\subset \mathcal {O}_S(X)$ equipped with the induced metric. These are called level sets of the warped cone. Further, we will denote by $\mathcal {O}_S^{[a,b]}(X)$ the subset $X\times [a,b]\subseteq \mathcal {O}_S(X)$ with the induced metric.

Given any sensible choice of the conical metric $d_\mathrm{cone}$, the level sets $\mathcal {O}_S^t(X)$ and the spaces are uniformly quasi-isometric (see e.g. [32]). For this reason we tend to confound them and call both of them ‘level sets’.

As in the the last few sections, we still assume the space X to be a ‘nice’ compact space and to be jumping-geodesic. For any fixed $\theta \ge 1$, it is easy to show that for $t\gg 0$ large enough . Moreover it is also simple to prove the following lemma:

Lemma 71

For every $\theta \ge 1$ there exists a $t_0$ large enough so that for every $t_0\le a \le t\le b\le \infty$ the natural inclusion and projection

induce isomorphisms

This Lemma allows us to mimic the proof of Theorem 55 in the context of (unified) warped cones.

Theorem 72

If has stable discrete fundamental group and $\mathcal {O}_S(X)$ is quasi-isometric to $\mathcal {O}_T(Y)$ then has stable discrete fundamental group and .

Proof (Sketch of proof)

Let $f:\mathcal {O}_S(X)\rightarrow \mathcal {O}_T(Y)$ be an (L, A)-quasi-isometry and let ${\bar{f}}$ be the coarse inverse. Also, fix three parameters $\theta ,\theta '$ and $\theta ''$ satisfying $\theta \ge L+A$, $\theta '\ge L\theta +A$ and $\theta ''\ge L(L\theta '+A)+A$ with $\theta$ large enough so that the projection is an isomorphism.

For every $a\gg 1$ there exist $c,b\gg 1$ such that

By Lemma 71, we can deduce that both

and

are surjective. Indeed, every $\theta '$-path Z in $\mathcal {O}_T^{[b,\infty ]}(Y)$ is equivalent to a 1-path in a level set which is sufficiently high up so that its image under ${\bar{f}}$ is a $\theta$-path in $\mathcal {O}_S^{[c,\infty ]}(X)$. This $\theta$-path is then is mapped to [Z] by $f_*$. The same argument works for ${\bar{f}}_*$ as well.

We can now find parameters $a>a'>a''\gg 1$ and $b>b'\gg 1$ so that the following composition of maps makes sense and it induces a commutative diagram:

To conclude, note that the dashed homomorphisms are induced by functions that are close to the identity and that $\iota _*$ and $p_*$ are isomorphisms. Then observe that Lemma 21 implies that the maps $f_*$ are also injective and hence all the maps are isomorphisms.