Topological Models of Abstract Commensurators

The full solenoid over a topological space $X$ is the inverse limit of all finite covers. When $X$ is a compact Hausdorff space admitting a locally path connected universal cover, we relate the pointed homotopy equivalences of the full solenoid to the abstract commensurator of the fundamental group $\pi_1(X)$. The relationship is an isomorphism when $X$ is an aspherical CW complex. If $X$ is additionally a geodesic metric space and $\pi_1(X)$ is residually finite, we show that this topological model is compatible with the realization of the abstract commensurator as a subgroup of the quasi-isometry group of $\pi_1(X)$. This is a general topological analogue of work of Biswas, Nag, Odden, Sullivan, and others on the universal hyperbolic solenoid, the full solenoid over a closed surface of genus at least two.


Introduction
The abstract commensurator of a group G, denoted Comm(G), is the collection of isomorphisms φ : H → K between finite-index subgroups H, K ≤ G, modulo equivalence which identifies isomorphisms agreeing on a finite-index domain.When G is infinite, Comm(G) is a natural relaxation of the automorphism group Aut(G), but is often more difficult to study.For example, it is not known whether Comm(F 2 ) is simple, where F 2 is a two-generated free group.Our goal is to provide topological and metric perspectives on Comm(G) parallel to those used to study Aut(G).
Given a group G with Eilenberg-MacLane space X = K(G, 1), the automorphism group Aut(G) is topologically modelled by E(X, * ), the group of pointed homotopy classes of pointed homotopy equivalences of X.When G is finitely generated, Aut(G) is metrically modelled by a natural map to QI(G), the quasi-isometry group of G.If X is a compact geodesic metric space, these two models of Aut(G) are compatible: each pointed homotopy equivalence of X lifts to a proper homotopy equivalence of the universal cover X preserving the G-orbit of the basepoint, which determines an element of QI(G) by the Milnor-Schwarz lemma.
The metric model of Aut(G) extends to a natural map Comm(G) → QI(G), a metric model of Comm(G).This article develops a topological model of Comm(G) compatible with this metric model and analogous to the homotopy model of Aut(G).
1.1.Statement of results.Our results apply to fundamental groups of compact Hausdorff spaces admitting a locally path-connected universal cover; spaces satisfying the latter condition are called unloopable.Following classic constructions of McCord [20], for any such space (X, * ) we construct the full solenoid over X as the inverse limit ( X, * ) of all finite-sheeted covers of (X, * ).
Theorem (3).Suppose (X, * ) is an unloopable pointed compact Hausdorff space.Then there is a homomorphism If X is homotopy equivalent to an aspherical CW complex, this map is an isomorphism.
2020 Mathematics Subject Classification.Primary 57M07; Secondary 20F65, 20F67, 57M10.EB was supported by the Azrieli Foundation.DS was supported in part by NSF grant DMS-1547292.
For example, let X = T n be an n-dimensional torus with G = π 1 (X, * ) ∼ = Z n .It is an exercise to prove that Comm(Z n ) ∼ = GL n (Q), and so there is an isomorphism E T n , * ∼ = GL n (Q).
After discussing abstract commensurators in Section 3 and reviewing the structure of full solenoids in Section 4, we prove Theorem 3 in Section 5.The proof combines the structure of the full solenoid as a fiber bundle over X with totally disconnected fibers with standard facts in shape theory applied to the inverse system of covering spaces used to define X.We present the arguments with care, starting with a review of inverse systems in Section 2, because the techniques may be unfamiliar to some readers.
Continuing the analogy with the automorphism group, for a residually finite group G we connect the topological model of Comm(G) to its realization in QI(G) in Section 7. To do so, we assume the base space X is a geodesic metric space and then use the metric structure on X as a space foliated by leaves quasi-isometric to G, which we develop in Section 6.
We were motivated to study full solenoids by work of Sullivan, with Biswas and Nag, who initiated the study of the universal hyperbolic solenoid Σ [2,26,27], the full solenoid over a closed surface Σ of genus g ≥ 2. Odden then proved that the pointed mapping class group Homeo( Σ, * )/ Homeo 0 ( Σ, * ) ∼ = Comm(π 1 (Σ)) provides a topological model of the abstract commensurator of the fundamental group of a closed surface [21].Odden connects the homeomorphisms of the solenoid to the action of Comm(π 1 (Σ)) on the boundary of the hyperbolic plane.Section 8 records a generalization in the setting of hyperbolic groups, relating the action of Comm(G) on ∂G to the solenoid model.
Theorem (Corollary 3).Suppose G ∼ = π 1 (X, * ), where (X, * ) is an unloopable compact geodesic metric space homotopy equivalent to an aspherical CW complex.Suppose G is a residually finite, torsion-free, non-elementary hyperbolic group.Then the homomorphism Q : E( X, * ) → QI(G) of Theorem 4 induces isomorphisms 1.2.Relationship with known commensurator computations.Corollary 3 fits into a familiar scheme of understanding Comm(G) by finding a map Comm(G) → Aut(X) for some object X on which G acts faithfully, then proving the map is an isomorphism onto the relative commensurator Comm Aut(X) (G).When G is a sufficiently nice branch group acting on a tree T , such as Grigorchuk's group of intermediate growth, Röver [23] gave an isomorphism Comm(G) ∼ = Comm Homeo(∂T ) (G).When G = F is Thompson's group, Burillo, Cleary, and Röver [5] , where PL ± 2 (R) is a certain group of piecewise linear homeomorphisms of R. When G = Mod(Σ) is the extended mapping class group of a closed surface of genus g ≥ 3 and C(Σ) is the curve complex of Σ, Ivanov [17] gave an isomorphism Comm(Mod(Σ)) ∼ = Comm Aut(C(Σ)) (Mod(Σ)).This has been generalized to other subgroups of mapping class groups, notably in recent work of Brendle and Margalit [3].
In each of these cases, structure of Aut(X) has allowed for more information about Comm(G) to be determined.Obtaining the strongest possible conclusion, Ivanov proved a rigidity statement Aut(C(Σ)) ∼ = Mod(Σ), which has the consequence that the natural map Mod(Σ) → Comm(Mod(Σ)) is an isomorphism.1.3.Bibliographic remarks.Solenoids are well-studied objects in classical topology, as a rich source of examples and counterexamples, as well as in dynamics, where they arise naturally in the study of foliations and actions on Cantor sets.Their appearance in dynamics goes back to work of Smale [24] and Williams [29], who studied solenoids arising as the inverse limit of iterated self-immersions of branched manifolds.
McCord's work, summarized in Theorem 1 in Section 4, identifies X with the suspension of the action of π 1 (X, * ) on its profinite completion by right translation.This is an example of a minimal, equicontinuous group action on a Cantor set.From this point of view, the full solenoid is an example of a foliated space called a matchbox manifold.Such dynamical systems have been studied systematically by Clark, Dyer, Hurder, and Lukina [7,8,11].
The structure of X as a principal bundle over X, which is used in the proof of Theorem 3, has been further studied by Clark and Fokkink [6] in the case that X is a closed manifold.Weak solenoids over closed manifolds were shown by Fokkink and Oversteegen to be fiber bundles with profinite group fiber [13, §7].

Inverse systems and pro-categories
Solenoids are by definition inverse limits of certain topological spaces.In order to relate their topology to the abstract commensurator we start by recalling the relevant categorical framework.We follow the language and conventions of Mardešić and Segal [19].
Given a category C an inverse system in C consists of a directed set Λ, a collection of objects {X λ } λ∈Λ and bonding morphisms p λλ ′ : To ease the burden of notation, when referring to the entire system we will use boldface without indexing information when such information is not needed.
Observe that there is a functor from C to pro-C that sends each object X to the rudimentary system (X) indexed by a singleton and that this allows us to treat C as a full subcategory of pro-C.In the interests of easing notation we will typically omit the system notation from rudimentary systems.
If (Λ, ≤) is a directed set and M ⊆ Λ, say that M is cofinal if, for every λ ∈ Λ, there is some µ ∈ M such that λ ≤ µ.Given any inverse system X = ({X λ }, p λλ ′ , Λ) in pro-C and directed subset M ⊆ Λ, then X M = ({X µ }, p µµ ′ , M ) is an inverse system and there is a restriction map of inverse systems i : We use the following notation for categories.Top is the category of topological spaces and continuous maps, and Cpt is the category of compact Hausdorff spaces and continuous maps.HTop is the category whose objects are topological spaces and whose morphisms are homotopy classes of continuous maps.HCpt is defined analogously.We use Top * , Cpt * , HTop * , and HCpt * to denote the respective pointed categories, in which each space is equipped with a basepoint, and all continuous maps and homotopies preserve basepoints.Grp is the category of groups with homomorphisms.The notation C(A, B) is used for the set of C morphisms from A to B.

Abstract commensurators
Write H ≤ f G to mean that H is a finite-index subgroup of a group G, and H f G to mean moreover that H is normal.A partial automorphism of a group G is an isomorphism between two finite index subgroups of G.If φ 1 : The abstract commensurator of a group G is the group For any subgroup H ≤ G, the relative commensurator of H in G is defined by For any group G, there is a homomorphism G → Aut(G) defined by g → c g , where The proof of the following lemma is straightforward.
Lemma 3.For any group G, 3.1.Commensurations as pro-automorphisms.For any abstract group G, let F (G) denote the collection of finite-index subgroups of G, equipped with inclusions The intersection of any two finite-index subgroups has finite index in G, so F (G) is the collection of groups of an inverse system G in the category Grp when ordered by reverse inclusion.
Proposition 1.Let G be a group and G be the inverse system of finite-index subgroups of G. Then there is an isomorphism Proof.For any H ≤ f G, the system of finite-index subgroups of H is cofinal in the system of finite-index subgroups of G.By Lemma 1, the inverse of the restriction map is a pro-Grp isomorphism i GH : It is an exercise to verify two properties.First, the assignment φ → φ * is functorial in the sense that (φ To see that ζ is well-defined, suppose φ : H → K is an isomorphism and ψ : H ′ → K ′ is the restriction of φ to some H ′ ≤ f H. Then by the above observations we have

GH
The fact that ζ is a homomorphism follows from functoriality of the assignment φ → φ * .
To see that ζ is injective, suppose a partial automorphism φ : GH and i −1 GK are restriction maps, by definition of equivalence there is a finite-index subgroup GH be the endomorphism of the inverse system F (G), with associated indexing function Combining the above observations we have f λ D = f ′ λ D , which completes the proof.

Full solenoids over unloopable spaces
The limit space of a sequence of finite-sheeted, regular covering maps appeared early in the study of homogeneous topological spaces as a source of examples and non-examples.McCord [20] gave a general account of the structure of such a space, not requiring coverings to be finite-sheeted, which he called a solenoidal space.Elsewhere in the literature these spaces have simply been called solenoids, while some authors reserve the name solenoid for inverse limits of systems of finitesheeted coverings of closed manifolds, or of only the circle (the original solenoids of van Dantzig and Vietoris [9,28]).Limits of sequences of finite-sheeted covers that are not necessarily regular are often called weak solenoids [22].We are interested in the limit of the inverse system of all finite-sheeted covers of a space, which we call the full solenoid (see Definition 3).After introducing the definitions with care we review McCord's work in this setting.Definition 2. A topological space X is unloopable if it is path-connected, locally path-connected, and semi-locally simply connected. 1Equivalently, X is unloopable if it has a locally path-connected universal cover.
We will typically work in categories whose objects are pointed topological spaces.When there is no chance for confusion, we will use the same symbol to denote the basepoints in different spaces.
Convention.When working with an unloopable pointed topological space (X, * ), we fix a pointed universal cover ( X, * ) → (X, * ) once and for all.There is a left action π 1 (X, * ) × X → X written (g, x) → gx.We then realize the Galois correspondence between connected covers of (X, * ) and subgroups of its fundamental group by setting (X H , * ) = H\( X, * ) for a subgroup H ≤ π 1 (X, * ).We will refer to constructions as canonical if they are canonical for a given choice of universal cover. 1 McCord calls such spaces nice [20].We borrow our terminology from Bourbaki, which uses the term dèlaçables.Definition 3. Given an unloopable pointed topological space (X, * ), the full solenoid over (X, * ), denoted ( X, * ), is the inverse limit of the system of connected pointed finite-sheeted covers of (X, * ) in Top * : The path-components of ( X, * ) are called leaves.
Given an unloopable pointed space (X, * ) with full solenoid ( X, * ), let (X, * ) denote the inverse system of finite-sheeted covers of (X, * ) with respect to a given universal cover.For any finite-index subgroup H ≤ f π 1 (X, * ) there is a projection map p H : ( X, * ) → (X H , * ).The collection of all such p H determines a map of inverse systems, hence a morphism p : ( X, * ) → (X, * ) in pro-Top * .
The construction of ( X, * ) is independent of choice of universal cover in the following senses.First, any two universal covers of an unloopable pointed space (X, * ) are homeomorphic by a unique morphism of pointed covering spaces.Any such homeomorphism induces homeomorphisms between covering spaces (X H , * ) compatible with the covering maps.It follows that ( X, * ) is well-defined up to homeomorphism commuting with the projection ( X, * ) → (X, * ).Second, for a given choice of universal cover, the collection {(X H , * ) | H ≤ f π 1 (X, * )} contains exactly one representative from each equivalence class of connected, finite-sheeted pointed cover of (X, * ).Therefore the full solenoid ( X, * ) is homeomorphic to the inverse limit of the collection of all connected, finite-sheeted pointed covers of (X, * ).
McCord [20] described the structure of inverse limits of sequences of regular covers of an unloopable base space.The results of his original paper apply to inverse limits of general directed systems of regular covers of an unloopable base space with essentially no change to the arguments.The system of finite-sheeted, regular covers of (X, * ) is cofinal in the system of finite-sheeted covers of (X, * ), so McCord's results apply to full solenoids over unloopable spaces.Definition 4. Given an unloopable pointed topological space (X, * ), the baseleaf of ( X, * ) is the image of the canonical map ℓ : ( X, * ) → ( X, * ) induced by the covering maps X → X H for H ≤ f π 1 (X, * ).
(1) There is a homeomorphism The canonical baseleaf map ℓ : X → X is the composition of X → {id} × X with the quotient.(2) Let K = ℓ( X) be the baseleaf.A subset of X is a path component if and only if it is of the form γK for some γ ∈ π 1 (X, * ).
Corollary 1. Suppose (X, * ) is an unloopable pointed topological space with residually finite fundamental group.The baseleaf map ℓ : X → X is injective with dense image.

The shape of solenoids
Our first main result, Theorem 3, is proved using the language of shape theory.The natural approach to shape for this setting is that of inverse systems, as described by Mardešić and Segal [19].The inverse system approach to shape associates certain inverse systems, known as expansions, to an object in a category.
Definition 5 ([19, Ch I. §2.1, p. 19]).Given a category T and an object X in T , a pro-T morphism p : X → X is a T -expansion 2 of X if X is an object of pro-T and p satisfies a universal property: for any pro-T morphism h : X → Y with Y in pro-T there is a unique pro-T morphism f making the following diagram commute: As we are primarily interested in morphisms, the following representation property of expansions is central.It follows from uniqueness that the function is natural in the sense that it respects composition; if In general, though, the assignment may be neither injective nor surjective [19, Ch I. §2.3, Remark 9].Specializing to full solenoids over unloopable pointed compact Hausdorff spaces, we are able to apply these facts to study self-homotopy equivalences in light of the following theorem.
The proof of Theorem 3 relies on the following proposition.Compare, for example, to improvability results of Geoghegan and Krasinkiewicz [14].Proposition 2. Suppose (X, * ) and (Y, ⋆) are unloopable pointed compact Hausdorff spaces with full solenoids ( X, * ) and ( Y , ⋆).Let (X, * ) and (Y, ⋆) be the inverse systems of connected, finite-sheeted pointed covers.Then there is a natural bijection HCpt * ( X, Y ) → pro-HCpt * (X, Y) Lemma 4. Suppose (X, * ) and (Y, ⋆) are unloopable pointed compact Hausdorff spaces.Let (X, * ) and (Y, ⋆) be the inverse systems of connected, finite-sheeted pointed covers.Then for every morphism [f ] ∈ pro-HCpt * (X, Y) there is a representative g that has a limit in Cpt * Proof.Let the inverse system X be indexed by Λ with bonding morphisms p λ1λ2 : X λ1 → X λ2 , and let Y be indexed by M with bonding morphisms q µ1µ2 : Y µ1 → Y µ2 .Both Λ and M are cofinite directed sets with unique minimum elements.Let m ∈ M denote the minimum index, so that Y m = Y .
By Lemma 2, any given morphism [f ] ∈ pro-HCpt * (X, Y) is represented by an order-preserving function φ : M → Λ and a collection of continuous maps By construction, for any µ ∈ M we we have pointed homotopies The homotopy lifting property provides a pointed homotopy f µ ∼ g µ .It follows that the collection of maps (g µ ) defines a map g of inverse systems in HCpt * and In fact, it follows from uniqueness of liftings that we may consider g as a morphism of systems in Cpt * .To see this, consider indices µ ≤ µ ′ ∈ M .Then while on the other hand Then uniqueness of lifts implies For each λ ∈ Λ, let p λ : X → X λ be the the system of projections.For each µ, define g µ : X → Y µ by g µ = g φ(µ) • p φ(µ) .These determine a continuous function g : X → Y , the limit of g.
Proof of Proposition 2. By Theorem 2, both ( X, * ) → (X, * ) and ( Y , ⋆) → (Y, ⋆) are HTop * -expansions.It follows from Fact 1 that there is a natural function Because (X, * ) and (Y, ⋆) are systems of compact Hausdorff spaces and HCpt * is a full subcategory of HTop * , this determines a natural function By Lemma 4, this map is surjective.Now suppose f, g : ( X, * ) → ( Y , ⋆) are pointed continuous maps whose pointed homotopy classes map to the same element [h] of pro-HCpt * (X, Y).Let [q] : Y → Y be the expansion morphism.Since f and g both map to h, by Fact 1, the morphisms q • f and q • g are representatives of the same element of pro-HCpt * ( X, Y).Consequently the maps q • f and q • g are pointed homotopic as maps ( X, * ) → (Y, ⋆), where q : Y → Y is the projection.
By Theorem 1, the map q is a fiber bundle over a compact Hausdorff space and therefore a fibration [25, §2.7, Cor 14].Therefore the pointed homotopy q • f ∼ q • g lifts to a homotopy f ∼ F for some F : X → Y .Because fibers of q are totally disconnected, q has the unique path lifting property [25, §2.2, Thm 5].This implies that both F and the homotopy f ∼ F are pointed.Moreover, because the baseleaf of X is path-connected, F and g agree on the baseleaf of X.Indeed, if γ is a based path of X starting at the basepoint, then q • F (γ) = q • g(γ), so that F (γ) is the unique lift of q • g(γ), and therefore equal to g(γ).But the baseleaf is dense in X, so we conclude that F = g as pointed continuous maps.
Combining these facts we arrive at a topological description of the abstract commensurator of π 1 (X, * ).For any pointed topological space (Y, ⋆), let E(Y, ⋆) be the automorphism group of (Y, ⋆) in Top * .Elements of E(Y, ⋆) are equivalence classes of pointed homotopy equivalences, modulo pointed homotopy.Theorem 3. Suppose (X, * ) is an unloopable pointed compact Hausdorff space.Then there is a homomorphism If X is homotopy equivalent to an aspherical CW complex, this map is an isomorphism.
Proof.Let (X, * ) be the full system of finite-sheeted pointed covers of (X, * ).By Proposition 2, there is a natural bijection HCpt * ( X, X) → pro-HCpt * (X, X) A standard characterization of pro-morphisms is the natural bijection [19, Ch I.
When X is homotopy equivalent to an aspherical CW complex, each X α is also homotopy equivalent to an aspherical CW complex, and so this function is a termby-term bijection by the Whitehead theorem.
Let π 1 (X, * ) be the pro-group obtained by applying the π 1 functor to the inverse system (X, * ).As noted above, there is a natural bijection Combining these natural morphisms, restricting to the automorphism groups in each category, and applying Proposition 1, we arrive at a homomorphism of groups that is an isomorphism when X is homotopy equivalent to an aspherical CW complex.

Metric notions
We now turn to the task of relating the topology of the full solenoid ( X, * ) to the geometry of the group π 1 (X, * ) in the case that X is an unloopable compact geodesic metric space and π 1 (X, * ) is residually finite.In this section we summarize the basic results used to metrize X in this case.
6.1.Metric spaces.When X is an unloopable length metric space with metric d X , X determines a metric on X, which will also be denoted d X .The metric on X is also a length metric [4,Proposition I.3.25], and the covering projection is a local isometry with respect to this metric.If X is additionally compact, then it follows from the metric Hopf-Rinow theorem [4, Proposition I.3.7] that both X and X are proper geodesic metric spaces.Moreover π 1 (X) is finitely presented and every finitely presented group arises this way [4,Corollary I.8.11].6.2.Quasi-isometries.For metric spaces (X, d X ) and (Y, d Y ), a map f : X → Y is a quasi-isometry if there are constants K ≥ 1 and C > 0 such that for any x 1 , x 2 ∈ X, and for any y ∈ Y there is some x ∈ X such that d Y (f (x), y) ≤ C. The quasi-isometry group of X is the group QI(X) of equivalence classes of quasi-isometries f : X → X, where f 1 and f 2 are equivalent if there is some If G is a finitely generated group, let QI(G) be the quasi-isometry group of G with respect to the word metric of some finite generating set.The quasi-isometry group is independent of chosen finite generating set.There is a natural map Comm(G) → QI(G), where [φ] ∈ Comm(G) determines a quasi-isometry of G by precomposing with any closest-point projection from G to the domain of φ.Whyte proved that this is injective, as recorded by Farb and Mosher [12, Proposition 7.5].
Lemma 5.For any unloopable compact geodesic metric space (X, * ) with fundamental group G = π 1 (X, * ), the orbit map G → ( X, * ) defined by g → g * induces an isomorphism QI(G) ∼ = QI( X).In order to relate the topology of the full solenoid ( X, * ) over (X, * ) to the coarse geometry of G = π 1 (X, * ) in Section 7, we will further require that the baseleaf map ℓ : X → X be an inclusion.By Corollary 1, this is equivalent to requiring that G be residually finite.

Profinite completions.
When a group G is finitely generated, for each n ≥ 1 there is a finite-index characteristic subgroup The subgroups G ≤n are a neighborhood basis of the identity for the topology on G induced by the pseudometric d defined by if such a maximum exists, and d(g residually finite then d is an ultrametric and the induced topology is Hausdorff.In this case, the profinite completion G is the metric completion of G with respect to d [15, Theorem 3.5], the natural homomorphism ι : G → G is an inclusion, and G acts freely by isometries on G by both left and right multiplication.6.4.Solenoid metric.Now suppose X is an unloopable compact geodesic metric space with residually finite fundamental group G = π 1 (X, * ).Then G is finitely presented, so by the above discussion there is a profinite metric d on G and and a proper, geodesic metric d X on X. Equip G × X with the ℓ ∞ product metric Both the action of G on ( G, d) by right multiplication and the left action of G on ( X, d X ) are isometric.Hence the left action of G on G × X defined in Section 4 is isometric with respect to d ∞ .Definition 6. Suppose X is an unloopable compact geodesic metric space with residually finite fundamental group G = π 1 (X, * ).The solenoid metric σ on X is the metric induced from the quotient metric on G × G X, obtained by taking the infimum of d ∞ over orbit representatives, under the identification X ≈ G × G X.
6.5.Metric balls in a full solenoid.Continue to consider an unloopable compact geodesic metric space X with residually finite fundamental group G = π 1 (X, * ).Definition 7. The injectivity radius of an unloopable metric space (Y, d Y ), denoted inj.rad(Y ), is the supremum of all R such that for all x ∈ Y the covering projection of B dY (x, R) is an isometry onto its image.
To see that the inclusion B → X is isometric onto its image, it suffices to show that if y, z ∈ B dX (x, ǫ) and g ∈ G, then d X (y, gz) ≥ 2ǫ.This follows from the reverse triangle inequality:

Homotopy and baseleaf quasi-isometry
In general, compactness improves the behavior of continuous functions between metric spaces.In the setting of full solenoids over spaces with residually finite fundamental group, compactness allows one to translate from the topology of the solenoid to the coarse geometry of the baseleaf.This has been studied in the context of foliated manifolds, for example in work of Hurder [16, §2.4].
Given an unloopable pointed space (X, * ), any pointed continuous map f : ( X, * ) → ( X, * ) maps the baseleaf into itself.When π 1 (X, * ) is residually finite, the baseleaf map ℓ : X → X is a bijection onto its image by Corollary 1. Identifying X with its image in X, say that f restricts to a function f X : ( X, * ) → ( X, * ).Lemma 7. Suppose (X, * ) is an unloopable compact geodesic metric space with residually finite fundamental group G = π 1 (X, * ).Suppose f : ( X, * ) → ( X, * ) is a pointed continuous function.Then for any C > 0 there exists Proof.Since X is compact and Hausdorff, G × G X is compact.Therefore, since f is continuous it is uniformly continuous in the σ metric on X.
Therefore, for any p ∈ X, repeated application of the triangle inequality gives Let HE( X, * ) be the collection of all pointed homotopy equivalences f : ( X, * ) → ( X, * ).Note that HE( X, * ) is a monoid, and E( X, * ) is the quotient group consisting of pointed homotopy classes.
Corollary 2. Given f, g ∈ HE( X, * ) such that g is a homotopy inverse of f , there exists an C such that for all p ∈ X, Combining this corollary with Lemma 7 we arrive at an exact description the coarse behavior of pointed homotopy equivalences.Lemma 9. Suppose (X, * ) is an unloopable compact geodesic metric space with residually finite fundamental group.For each f ∈ HE( X, * ), the restriction f X is a quasi-isometry of ( X, d X ).
Proof.Given any f ∈ HE( X, * ), fix a homotopy inverse g.By Corollary 2 there exists a C such that for all p ∈ X, we have d Next, by Lemma 7 with L = max{ Lip C (f ), Lip C (g)}, both f and g are (L, C)coarse Lipschitz on ( X, d X ).This is a characterization of an (L, C)-quasi-isometry of ( X, d X ) [10, Corollary 8.13] and we are done.Proposition 3. Suppose (X, * ) is an unloopable compact geodesic metric space with residually finite fundamental group G = π 1 (X, * ).There is a group homomorphism given by [f ] → f X for a representative f ∈ HE( X, * ).
Proof.By Lemma 9, given f ∈ HE( X, * ), the restriction f X is a quasi-isometry of X.The assignment f → f X a monoid homomorphism HE( X, * ) → QI( X) Next, if f, g ∈ HE( X, * ) and f ∼ g, then by Lemma 8 there exists a constant C such that for all p ∈ X, the distance d X (f (p), g(p)) ≤ C. Thus f X and g X represent the same element of QI( X).Therefore we obtain a monoid homomorphism Every monoid homomorphism of groups is a group homomorphism.The conclusion follows by composing with the isomorphism QI( X) ∼ = QI(G) of Lemma 5. Proof.Suppose [f ] ∈ E( X, * ).By Lemma 4, there is an morphism of the covering system {f α : X φ(α) → X α } with limit f homotopic to f .As in the proof of Theorem 3, the collection of induced maps (f α * ) is a pro-Grp automorphism of the inverse system of finite-index subgroups of G.
Pick an index α and set H 0 = π 1 (X φ(α) , * ).As in the proof of Proposition 1, there are finite-index subgroups The inclusion of [φ] into QI(G) is the composition of nearest point projection to H and φ : H → K. Let ( X, p) → (X, * ) be the universal cover of X.Under the identification QI(G) ∼ = QI( X) induced by the orbit map g → gp, the image of [φ] in QI( X) is the map hp → φ(h)p, which bijectively maps the H-orbit of the basepoint p to the K-orbit of p, precomposed with a closest-point projection X → Hp.
Let f α : ( X, p) → ( X, p) be the unique lift of f α .Since f is the limit of f α , for any point hp in the H-orbit of the basepoint p ∈ X we have Since H and K are finite-index subgroups of G this implies f is bounded distance from the image of φ in QI( X).Therefore, by Lemma 8, f and φ are bounded distance maps, and we are done.

Hyperbolicity and the boundary realization
For a closed hyperbolic surface Σ, Odden proved the existence of an isomorphism Comm(π 1 (Σ, * )) ∼ = Homeo( Σ, * )/ Homeo 0 ( Σ, * ) by proving that both groups are isomorphic to the relative commensurator of π 1 (Σ) in Homeo(∂H 2 ) [21, Theorems 4.6, 4.12].We will prove that the abstract commensurator of any torsion-free, non-elementary hyperbolic group G is isomorphic to its relative commensurator in the group of homeomorphisms of its boundary, generalizing half of Odden's proof.We then provide a topological analogue of the other half of the proof in the case G acts on a sufficiently nice metric space X homotopy equivalent to a finite aspherical CW complex.
Suppose G is a Gromov hyperbolic group.Its boundary ∂G is well-defined up to G-equivariant homeomorphism.There is a well-defined map QI(G) → Homeo(∂G), which is injective in the case |∂G| > 3 [10,Corollary 11.115]A group G has the unique root property if g n = h n implies g = h for all g, h ∈ G and all nonzero n ∈ Z.If G has the unique root property and φ 1 : H 1 → K 1 and φ 2 : H 2 → K 2 are equivalent partial automorphisms of G, then φ 1 and φ 2 agree on Lemma 2.4].Note that if G satisfies the unique root property then for any H ≤ f G the natural map Aut(H) → Comm(G) is injective.
We now appeal to basic facts about the action of a hyperbolic group on its boundary-see the survey of Kapovich and Benakli for a reference [18, §4].Every g ∈ G has exactly two fixed points on ∂G, an attracting fixed point g + and a repelling fixed point g − , and attracting fixed points of elements of G are dense in ∂G.Suppose x ∈ ∂G is an attracting fixed point g + for g ∈ G. Replacing g with a sufficiently large positive power, we may assuming g ∈ G 1 .It follows from (1) that f −1 ( f (g + )) is a fixed point of g with the same dynamical properties as g + , therefore f −1 ( f (g + )) = g + .By density of the set of attracting fixed points, this implies f = f .Corollary 3. Suppose (X, * ) is an unloopable compact geodesic metric space homotopy equivalent to an aspherical CW complex.Suppose G = π 1 (X, * ) is a residually finite, torsion-free, non-elementary hyperbolic group.Then the map Q : E( X, * ) → QI(G) of Proposition 3 induces an isomorphism E( X, * ) ∼ = Comm Homeo(∂G) (G).
Proof.By Theorem 4, the image of Q lies in Comm(G) ≤ QI(G).Because X is homotopy equivalent to an aspherical CW complex, Q defines an isomorphism E( X, * ) → Comm(G) by Theorem 3. The result follows from Theorem 5.

Fact 1 (
[19, Ch I.  §2.3, p. 25]).Given T -expansions p : X → X and q : Y → Y and a T morphism f : X → Y , there exists a unique pro-T morphism f such that the following square commutes in pro-T2 Mardešić and Segal also treat the more general notion of a P-expansion for a full subcategory P of T .We only need to consider T -expansions.