## BI: Finite decomposition complexity (is preserved by relative hyperbolicity)

In this post I’ll define finite decomposition complexity (FDC) for you, tell you what it’s good for (Stable Borel Conjecture!) and point out that an argument by Osin shows that it is preserved by relative hyperbolicity.

First, the motivation. Here is a very cool conjecture.

Borel Conjecture: If two closed aspherical manifolds have isomorphic fundamental groups, then they are homeomorphic.

A manifold $M$ is aspherical if its universal cover is contractible, so that $\pi_1(M)$ determines the homotopy type of $M$. In case you don’t see the point, let me tell you that the Borel Conjecture for $S^1\times S^1\times S^1$ implies the Poincaré Conjecture.
The conjecture has been established for high dimensional manifolds of non-positive curvature by Farrell and Jones, and several people have worked on it since then, most notably showing that the conjecture holds if the fundamental groups involved are hyperbolic or CAT(0).

There are several conjectures related to the Borel Conjecture, for example the Novikov Conjecture, and here is another one.

Stable Borel Conjecture (SBC): If the closed aspherical manifolds $M, M'$ have isomorphic fundamental groups, then there exists $n$ so that $M\times \mathbb{R}^n$ is homeomorphic to $M'\times \mathbb{R}^n$.

Stabilisation procedures appear all over topology. For example, there are many contractible 3-manifolds (without boundary) that are not homeomorphic to $\mathbb{R}^3$. However, when taking the product of such a manifold with $\mathbb{R}$  one actually gets $\mathbb{R}^4$. Just to say that the SBC is a natural weaker version of the full Borel Conjecture. Of course, $M$ is said to satisfy the SBC if the statement holds for the given $M$.

And here’s the motivation for being interested in FDC.

Theorem (Guentner, Tessera, Yu): If $\pi_1(M)$ has finite decomposition complexity, then $M$ satisfies the Stable Borel Conjecture.

(Guoliang Yu told me that one can take $n=3$ in statement of the SBC.)

Time for the definition of FDC. We say that $X=\bigcup X_\alpha$ is an $R$-decomposition of the metric space $X$ if $d(X_\alpha,X_\beta)>R$ for each $\alpha\neq \beta$. Let us consider the following decomposition game. Fix a metric space $X$. Bob’s aim is to find a nice decomposition of $X$, and Alice’s aim is to prevent this.

Step 0: Alice gives Bob a huge real number $R_0$. Bob finds an $R_0$-decomposition $\bigcup X^1_\alpha$ of $X$.

Inductive step: A family of subspaces $X^i_\alpha$, coming from the previous step, is given. Alice gives Bob a real number $R_i$ and Bob finds  an $R_i$-decomposition of each $X^i_\alpha$. $X^{i+1}_\alpha$ is the union of all subspaces appearing in each decomposition.

Definition: $X$ has FDC if Bob has a strategy to end up with a collection of uniformly bounded subspaces of $X$.

Having FDC is a quasi-isometry invariant, and actually a coarse invariant.

If you know what asymptotic dimension is, it’s a nice exercise to show that it implies FDC. Otherwise, try to do it for $\mathbb{R}^n$. Hint: find a nice coloured covering with finitely many subsets of diameter $\leq D$, for each $D$.
In particular, say, hyperbolic groups have FDC.

Not all groups with FDC have finite asymptotic dimension. Indeed, GTY showed that linear groups have FDC, but some of them contain an infinite sum of copies of $\mathbb{Z}$ and hence have infinite asymptotic dimension.

Another nice fact is that the class of groups with FDC is closed under extensions. Indeed, GTY observed that the following holds in the context of metric spaces. In order to state it, notice that it makes sense to play the decomposition game starting from a family of metric spaces, rather than one metric space.

Fibering Theorem: Suppose that $f:X\to Y$ is Lipschitz and that $Y$ as well as $\{f^{-1}(B_i)\}$ have FDC, for each family of uniformly bounded subsets $\{B_i\}$ of $Y$. Then $X$ has FDC.

A similar statement for asymptotic dimension is usually called Hurewicz Theorem (as it is inspired by an actual theorem of Hurewicz about topological dimension).
The proof is simple. Bob first pretends he’s playing the decomposition game on $Y$, and when he as won in $Y$ he’s left with preimages of bounded sets of $Y$ in $X$, where he also knows how to win.

Let me state another permanence result. It is not true that if a metric space is an infinite union of subspaces with FDC then the ambient space has FDC. However, the following result tells us that this is true if the subspaces can be “separated”. You may think of $X$ as obtained gluing metric spaces $\{X_i\}$ to some $Y_0$, and then you can let $Y_R$ be the $R/2$-neighborhood of $Y_0$.

Infinite Union Theorem: Suppose that $X=\bigcup X_i$ and that $\{X_i\}$ has FDC. Also, suppose that for every $R$ there exists $Y_R$ with FDC  so that $\{X_i\backslash Y_R\}$ are pairwise $R$-disjoint. Then $X$ has FDC.

Finally, as promised, I’ll tell you how one can show the following.

Theorem: A relatively hyperbolic group has FDC if and only if the peripherals do.

The only if part follows from the fact that having FDC is stable under taking subgroups.
For the if part, we can use arguments used by Denis Osin to show the analogue statement for finite asymptotic dimension.

What he did is considering the (Lipschitz) map $f$ from the relatively hyperbolic group $G$ to its coned off graph, $\hat{G}$, and showing that the hypothesis of Fibering/Hurewicz Theorem for asymptotic dimension apply to such map. In particular, he showed that $\hat{G}$ has finite asymptotic dimension, and so in particular it has FDC. To show that the Fibering Theorem applies we now have to look at preimages of balls in $\hat{G}$, and proceed by induction. Let $X_n$ be the preimage of the ball of radius $n$ in $\hat{G}$. Then $X_{n+1}$ is obtained as a union of $X_n$ and some cosets of peripheral subgroups (almost true). This is the perfect setting for using the Infinite Union Theorem, with $Y_R$ being suitable neighborhoods of $X_n$, and indeed Osin showed that the result applies. It shouldn’t be too hard to convince yourself of this fact in the case of a free product.

This was the proof outline, and one can actually check that Denis only used the Fibering Theorem and the Infinite Union Theorem (and properties of relatively hyperbolic groups), and hence his proof also gives permanence of FDC.

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