##
**Coarse differentiation of quasi-isometries. I: Spaces not quasi-isometric to Cayley graphs.**
*(English)*
Zbl 1264.22005

Given a metric space \(X\) denote by \(\operatorname{QI}(X)\) the group of quasi-isometries of \(X\) modulo those of bounded distance from the identity. The paper under review computes the group \(\operatorname{QI}(X)\) for various horocyclic products of negatively-curved spaces and applies these results to establish important new quasi-isometric rigidity results. The proofs are based on the method of coarse differentiation introduced by the authors in [Pure Appl. Math. Q. 3, No. 4, 927–947 (2007; Zbl 1167.22007)].

Recall that given two negatively-curved spaces \(X_1\), \(X_2\) with respective Busemann functions \(f_1\), \(f_2\), their horocyclic product is the subset \[ X := \{(x_1, x_2) \in X_1 \times X_2\mid f_1(x_1) = -f_2(x_2)\} \] of the product \(X_1 \times X_2\). Such spaces come equipped with a natural height function \(h := f_1 = -f_2\). The authors consider specifically the family of Diestel-Leader graphs DL\((m,n)\), which are horocyclic products of two regular trees of valencies \(m+1\) and \(n+1\), respectively, and a family of \(3\)-dimensional solvable Lie groups Sol\((m,n)\), which are isomorphic to horocyclic products of two planes of constant negative curvatures depending on the parameters \(m\) and \(n\).

Quasi-isometries of these spaces which coarsely preserve the height function have been classified by Farb and Mosher. They are all essentially equivalent to restrictions of products of quasi-isometries of the two factors, hence correspond to pairs of bi-Lipschitz maps of the corresponding pointed boundaries. The main technical result of the paper under review greatly improves on this result by showing that actually every quasi-isometry is at bounded distance from a coarsely height-preserving one, hence of the above product form (Theorem 2.1/Theorem 2.3). In the case of distinct curvatures \(m \neq n\), this implies \[ \operatorname{QI}(\operatorname{Sol}(m,n)) \cong \operatorname{Bilip}(\mathbb R)^2, \quad \operatorname{QI}(\operatorname{DL}(m,n)) \cong \operatorname{Bilip}(\mathbb Q_m) \times \operatorname{Bilip}(\mathbb Q_n), \] whereas, in the case of equal curvature \(m=n\), there is an additional \(\mathbb Z/2\mathbb Z\)-factor corresponding to the flip. Actually, only the distinct curvature case is completely established in the present paper, whereas parts of the proof in the case of equal curvature are deferred to a sequel.

The identifications of the above \(\operatorname{QI}\)-groups have a number of striking applications. Firstly, if \(m \neq n\), then neither \(\operatorname{Sol}(m,n)\) nor DL\((m,n)\) are quasi-isometric to Cayley graphs of finitely-generated groups (Theorem 1.2/Theorem 1.4), which is the result alluded to in the title. Secondly, the authors determine completely which of the spaces \(\operatorname{Sol}(m,n)\), respectively, \(\operatorname{DL}(m,n)\), are quasi-isometric to each other, assuming \(m \neq n\). For example, \(\operatorname{Sol}(m,n)\) is quasi-isometric to \(\operatorname{Sol}(m',n')\) if and only if \(m/m' = n/n'\) (Theorem 1.3). Both results have counterparts in the equal curvature case (the former in terms of a quasi-isometric rigidity statement), but these are deferred to the sequel. To deduce the above results from the knowledge of the QI-groups, the authors rely heavily on previous results concerning quasi-isometric quasi-actions on trees and hyperbolic spaces.

The starting point for the proof of the main technical result is a coarse version of Rademacher’s theorem (Theorem 4.3), in which the authors show that quasi-isometries of the spaces in question can be coarsely differentiated on a set of large measure. Using this theorem, the authors deduce that on a set of large measure any given quasi-isometry looks locally like standard product quasi-isometries. The main technical step (Step II) is to show that the local maps preserve the up-direction, which is necessary to assemble the local information. This is the only step where the difference between the distinct curvature case and the equal curvature case becomes relevant.

The results demonstrate the power of the method of coarse differentiation to establish new rigidity results, in particular for solvable groups. It would be interesting to know which of the arguments generalize to more general horocyclic products.

Recall that given two negatively-curved spaces \(X_1\), \(X_2\) with respective Busemann functions \(f_1\), \(f_2\), their horocyclic product is the subset \[ X := \{(x_1, x_2) \in X_1 \times X_2\mid f_1(x_1) = -f_2(x_2)\} \] of the product \(X_1 \times X_2\). Such spaces come equipped with a natural height function \(h := f_1 = -f_2\). The authors consider specifically the family of Diestel-Leader graphs DL\((m,n)\), which are horocyclic products of two regular trees of valencies \(m+1\) and \(n+1\), respectively, and a family of \(3\)-dimensional solvable Lie groups Sol\((m,n)\), which are isomorphic to horocyclic products of two planes of constant negative curvatures depending on the parameters \(m\) and \(n\).

Quasi-isometries of these spaces which coarsely preserve the height function have been classified by Farb and Mosher. They are all essentially equivalent to restrictions of products of quasi-isometries of the two factors, hence correspond to pairs of bi-Lipschitz maps of the corresponding pointed boundaries. The main technical result of the paper under review greatly improves on this result by showing that actually every quasi-isometry is at bounded distance from a coarsely height-preserving one, hence of the above product form (Theorem 2.1/Theorem 2.3). In the case of distinct curvatures \(m \neq n\), this implies \[ \operatorname{QI}(\operatorname{Sol}(m,n)) \cong \operatorname{Bilip}(\mathbb R)^2, \quad \operatorname{QI}(\operatorname{DL}(m,n)) \cong \operatorname{Bilip}(\mathbb Q_m) \times \operatorname{Bilip}(\mathbb Q_n), \] whereas, in the case of equal curvature \(m=n\), there is an additional \(\mathbb Z/2\mathbb Z\)-factor corresponding to the flip. Actually, only the distinct curvature case is completely established in the present paper, whereas parts of the proof in the case of equal curvature are deferred to a sequel.

The identifications of the above \(\operatorname{QI}\)-groups have a number of striking applications. Firstly, if \(m \neq n\), then neither \(\operatorname{Sol}(m,n)\) nor DL\((m,n)\) are quasi-isometric to Cayley graphs of finitely-generated groups (Theorem 1.2/Theorem 1.4), which is the result alluded to in the title. Secondly, the authors determine completely which of the spaces \(\operatorname{Sol}(m,n)\), respectively, \(\operatorname{DL}(m,n)\), are quasi-isometric to each other, assuming \(m \neq n\). For example, \(\operatorname{Sol}(m,n)\) is quasi-isometric to \(\operatorname{Sol}(m',n')\) if and only if \(m/m' = n/n'\) (Theorem 1.3). Both results have counterparts in the equal curvature case (the former in terms of a quasi-isometric rigidity statement), but these are deferred to the sequel. To deduce the above results from the knowledge of the QI-groups, the authors rely heavily on previous results concerning quasi-isometric quasi-actions on trees and hyperbolic spaces.

The starting point for the proof of the main technical result is a coarse version of Rademacher’s theorem (Theorem 4.3), in which the authors show that quasi-isometries of the spaces in question can be coarsely differentiated on a set of large measure. Using this theorem, the authors deduce that on a set of large measure any given quasi-isometry looks locally like standard product quasi-isometries. The main technical step (Step II) is to show that the local maps preserve the up-direction, which is necessary to assemble the local information. This is the only step where the difference between the distinct curvature case and the equal curvature case becomes relevant.

The results demonstrate the power of the method of coarse differentiation to establish new rigidity results, in particular for solvable groups. It would be interesting to know which of the arguments generalize to more general horocyclic products.

Reviewer: Tobias Hartnick (Haifa)

### MSC:

22E25 | Nilpotent and solvable Lie groups |

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

20F65 | Geometric group theory |

22E40 | Discrete subgroups of Lie groups |

53C24 | Rigidity results |

05C05 | Trees |

05C76 | Graph operations (line graphs, products, etc.) |

### Citations:

Zbl 1167.22007
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\textit{A. Eskin} et al., Ann. Math. (2) 176, No. 1, 221--260 (2012; Zbl 1264.22005)

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