Lifting Group Actions, Equivariant Towers and Subgroups of Non-positively Curved Groups
aa r X i v : . [ m a t h . G R ] M a r A lgebraic & G eometric T opology XX (20XX) 1001–999 Lifting Group Actions, Equivariant Towers and Subgroupsof Non-positively Curved Groups R ICHARD G AELAN H ANLON E DUARDO M ART ´ INEZ -P EDROZA If C is a class of complexes closed under taking full subcomplexes and covers and G is the class of groups admitting proper and cocompact actions on one-connectedcomplexes in C , then G is closed under taking finitely presented subgroups. Asa consequence the following classes of groups are closed under taking finitelypresented subgroups: groups acting geometrically on regular CAT(0) simplicialcomplexes of dimension 3, k –systolic groups for k ≥
6, and groups actinggeometrically on 2–dimensional negatively curved complexes. We also show thatthere is a finite non-positively curved cubical 3–complex which is not homotopyequivalent to a finite non-positively curved regular simplicial 3–complex. Weinclude applications to relatively hyperbolic groups and diagramatically reduciblegroups. The main result is obtained by developing a notion of equivariant towerswhich is of independent interest.20F67; 57M07
We show that some classes of non-positively curved groups are closed under takingfinitely presented subgroups. We assume all spaces are combinatorial complexes andall maps are combinatorial, see Section 3.1 for a definition. A complex is one-connected if it is connected and simply-connected. A subcomplex K of X is full if for any cell σ ⊂ X , ∂σ ⊂ K implies σ ⊂ K . An action of a group G on a space X is proper if for all compact subsets K of X there are finitely many group elements g such that K ∩ g ( K ) = ∅ . The action is cocompact if there is a compact subset K of X such thatthe collection { gK : g ∈ G } covers X . Our main result is the following. Theorem 1.1
Let C beacategoryofcomplexesclosedundertakingfullsubcomplexesand topological covers. Let G be the category of groups acting properly and cocom-pactly, by combinatorial automorphisms, on one-connected complexes in C . Then G isclosed under taking finitely presented subgroups. Published: XX Xxxember 20XX
DOI: 10.2140/agt.20XX.XX.1001
R.G. Hanlon and E. Mart´ınez-Pedroza
Some words about the literature. A theorem of Steve Gersten states that finitelypresented subgroups of hyperbolic groups of cohomological dimension ≤ A regular simplicial complex is a piecewise Euclidean simplicial complex where each1–cell has unit length. A result of Rena Levitt shows that the category of regularlocally CAT(0) simplicial complexes of dimension 3 is closed under taking full sub-complexes [15]. It is immediate that this category is closed under taking covers. Recallthat a group action on a metric space is said to be geometric if is proper, cocompactand by isometries. Corollary 1.2 If G acts geometrically on a regular CAT(0) simplicial complex ofdimension 3, then any finitely presented subgroup acts geometrically on a regularCAT(0) simplicial complex ofdimension 3.It is not known whether Corollary 1.2 holds for higher dimensions. However, the proofpresented here does not generalize since Levitt has exhibited regular locally CAT(0)simplicial complexes of dimension ≥ X such that π ( X ) contains a finitely presented subgroup which does not admita finite classifying space. Since every compact locally CAT(0) space is a classifyingspace for its fundamental group, the previous corollary together with Brady’s exampleimplies the following statement. Corollary 1.3
Thereisa 3–dimensionalfinitelocally CAT(0) cubicalcomplexwhichisnothomotopyequivalenttoa 3–dimensionalfiniteregularlocally CAT(0) simplicialcomplex. A lgebraic & G eometric T opology XX (20XX)quivariant Towers and Non-positively Curved Groups Levitt proved that non-positively regular CAT(0) simplicial complexes of dimension3 are systolic complexes [16], for this larger class our main theorem also applies.The notion of simplicial nonpositive curvature for simplicial complexes was introducedby Januszkiewicz and ´Swia¸tkowski in [14] and independently by Haglund [9] as acombinatorial analog of nonpositive curvature metric conditions. A simplicial complex L is flag if any set of vertices which are pairwise connected by 1–cells of L , spans asimplex in L . A simplicial complex L is k–large , k ≥
6, if L is flag and there areno embedded cycles of length < k which are full subcomplexes of L . A simplicialcomplex X is locally k–large if links of all simplices in X are k –large. The fact that thecategory of locally k –large simplicial complexes is closed under taking covers and full-subcomplexes immediately follows from the definitions. A group is k–systolic if it actsproperly and cocompactly by simplicial automorphisms on a one-connected locally k –large simplicial complex. Wise proved that finitely presented subgroups of torsion free k –systolic groups are k –systolic using a tower argument [23]. Corollary 1.4 extendsWise’s result to include groups with torsion. Previously to this work, the statement ofCorollary 1.4 was also proved by Gaˇsper Zadnik using different methods [25]. Corollary 1.4
For k ≥
6, if G is a k –systolic group then any finitely presentedsubgroup of G is k –systolic.In [19], Osajda introduced the notion of complexes with SD ∗ ( k ) links for k ≥
6. Thereis proved that the class of fundamental groups of compact complexes with SD ∗ ( k ) linksis closed under taking finitely presented subgroups [19, Thm. 8.7]. His proof is a towerargument for compact complexes, and in particular it is shown that SD ∗ ( k ) complexesare closed under taking covers and full subcomplexes. It follows that Theorem 1.1applies to the class of SD ∗ ( k ) providing an extension of Osadja’s result.A 2–complex X is negatively curved if it satisfies one of the following conditions:(1) (metric condition) there is κ < X admits the structure of a locallyCAT( κ ) M κ –complex, or(2) (conformal condition) there is an assignment of a non-negative real number,called an angle , to each corner of each 2–cell such that the sum of the angles onan n –gon is strictly less than ( n − π and links of 0–cells satisfy Gromov’s linkcondition : every non-trivial circuit in the link is of angular measure at least 2 π .A group acting geometrically on a one-connected negatively curved 2–complex is wordhyperbolic [3, 8]. Gersten proved that finitely presented subgroups of fundamentalgroups of finite negatively curved 2–complexes are word hyperbolic [7, Thm 2.1]. A lgebraic & G eometric T opology XX (20XX) R.G. Hanlon and E. Mart´ınez-Pedroza
One easily verifies that the category of negatively curved 2–complexes is closed undertaking subcomplexes and covers. The following corollary extends Gersten’s result.
Corollary 1.5 (Subgroups of 2–Diml Negatively Curved Groups) Let Y be a one-connectednegativelycurvedproperandcocompact H-2 –complex. If G ≤ H isfinitelypresented then G isword hyperbolic.We remark that Corollary 1.5 would follow from Gersten’s result if H is known to bevirtually torsion free. However, it is an open question whether hyperbolic groups arevirtually torsion free [8]. We also obtain an analogous result for relatively hyperbolicgroups stated below.A G –complex X is almost proper if G acts properly on the complement of the zero-skeleton of X . Observe that a proper action is almost proper. A group is called slender if all its subgroups are finitely generated. For definitions of relatively hyperbolic groupsand fine complexes we refer the reader to Section 5.2. Theorem 1.6 (Theorem 5.15) [Subgroups of 2–Diml Relatively Hyperbolic Groups]Let Y beaone-connected negatively curved, fine,almost proper and cocompact H-2 –complex such that H –stabilizers of cells are slender. If G ≤ H is finitely presentedthen G ishyperbolic relative toafinite collection of G –stabilizers ofcells of Y .The term of diagramatically reducible complex, defined below, was introduced byGersten in connection with the study of equations over groups [6]. The notion wasfirst used, with different names, by Chiswell, Collins and Huebschmann [4] and Sier-adski [22]. Recall that an immersion is a locally injective map, and a near-immersion is a map which is locally injective except at 0–cells of the domain. A 2–complex X is diagramatically reducible if there are no near-immersions C → X , where C is a cellstructure for the 2–dimensional sphere.Since the composition of a near-immersion followed by an immersion is a near-immersion, the category of diagramatically reducible complexes is closed under tak-ing covers and subcomplexes. This category of complexes includes locally CAT(0)2–complexes, certain classes of small cancellation complexes, conformal negativelycurved 2–complexes, spines of hyperbolic knots, and non-positively curved squarecomplexes to name a few examples. Recall that a proper G –complex X is a modelfor EG if for every finite subgroup F ≤ G , the fixed point set X F is contractible. Thefollowing theorem is proved in Section 5. A lgebraic & G eometric T opology XX (20XX)quivariant Towers and Non-positively Curved Groups Theorem 1.7 (Theorem 5.10) [Diagramatically Reduced Groups] Let X be a di-agramatically reducible one-connected proper H-2 –complex. If G ≤ H is finitelypresentedthen G admitsadiagramatically reducible 2–dimensional cocompactmodelfor EG . Briefly, Theorem 1.1 is proved using an extension of the tower method to equivariantmaps. There are several applications of towers in combinatorial group theory asillustrated in [7, 11, 12, 13, 23, 24] and the results of this paper. Towers is a geometrictechnique from 3–manifold topology introduced by Papakyriakopolous [20], and laterbrought to combinatorial group theory by Howie [12]. A combinatorial map X → Y between connected CW-complexes is a tower if it can be expressed as a compositionof inclusions and covering maps. A tower lifting f ′ of f is a factorization f = g ◦ f ′ where g is a tower. The lifting f ′ is trivial if g is an isomorphism and the lifting is maximal if the only tower lifting of f ′ is the trivial one. It is well known that if X isa finite complex, then any combinatorial map X → Y admits a maximal tower [11,Lem. 3.1]. A tower is called an F –tower if it is a composition of covering maps andinclusions of full-subcomplexes.By a locally finite complex we mean a complex such that every closed cell intersectsfinitely many closed cells. Theorem 1.8 (Theorem 3.14) [Maximal Equivariant F –Towers] Let f : X → Y bea G –map. If X is one-connected and G –cocompact and Y islocally finite,then f hasamaximal F –tower lifting f = g ◦ f ′ where g and f ′ are G –maps.An analogous result to Theorem 1.8 where Y is not required to be locally finite andinstead f ′ is only a maximal tower lifting (not a maximal F –tower lifting) also holds,see Theorem 3.18. This slightly different result is relevant to applications such asTheorem 1.6.In [23], there is a result similar to Theorem 1.8 stating that if X → Y is a map betweenfinite simplicial complexes then there is a maximal expanded -tower lifting. This is adifferent class of towers, and neither result subsume the other. The class of expandedtowers works well in the setting of 2–skeletons of systolic complexes which Wise usedto prove Corollary 1.4 in the torsion-free case.A consequence of Theorem 1.8 is the following. A lgebraic & G eometric T opology XX (20XX) R.G. Hanlon and E. Mart´ınez-Pedroza
Theorem 1.9 (Theorem 4.1) [Existence of Immersed Cocompact F –Cores] If Y isaone-connected,properandlocallyfinite H –complexand G ≤ H isfinitelypresented,thenthereisaone-connected cocompact G –complex X anda G –equivariant F –tower X → Y .The proof of Theorem 1.9 relies on the construction of a one-connected and cocompact G –complex X together with a G –map f : X → Y ; this construction uses the hypoth-esis that Y is one-connected. Given such a map, since X is one-connected, there is anequivariant maximal F –tower lifting f = g ◦ f ′ with g : X → Y . The maximality of f ′ implies that X is one-connected, and together with Y being locally finite, that X is G –cocompact.We also prove a version of Theorem 1.9 without the assumption that Y is locallyfinite, but with the weaker conclusion that the G –map X → Y is only a tower, seeTheorem 4.4. The proof of our main result, Theorem 1.1, follows immediately fromTheorem 1.9. Proof of Theorem 1.1
Let Y be a one-connected complex in C , let H be a groupacting properly and cocompactly on Y , and let G ≤ H be a finitely presented subgroup.Since Y is a proper and cocompact H –complex, it is locally finite. By Theorem 1.9,there is a one-connected cocompact G –complex X and an G –equivariant F –tower X → Y . By equivariance, X is also a proper G –complex. Since C is closed undertaking full-subcomplexes and covers, it follows that X is in C . Therefore G is in G . The rest of the paper is organized in four sections. Section 2 contains a result thatprovides sufficient conditions to lift a group action on a space to an intermediate cover.Section 3 contains the definition of equivariant towers and the proof of the existenceof maximal equivariant F –towers. Section 4 contains the proof of the existence ofimmersed cocompact F –cores. The last section contains the proofs of Theorems 1.5and 1.7. Acknowledgments
Thanks to Daniel Wise and Mark Sapir for both suggesting the applications to systoliccomplexes, and Daniel Wise for sharing his unpublished preprint [23] which motivated A lgebraic & G eometric T opology XX (20XX)quivariant Towers and Non-positively Curved Groups part of the work of this paper. We also thank Damian Osajda for useful comments. Wealso thank Saya Mart´ınez-Aoki for assistance during this work. Last but not least, wethank the anonymous referee for feedback, several suggestions and corrections, andpointing out a serious mistake in an earlier version of the paper. We also acknowledgefunding by the Natural Sciences and Engineering Research Council of Canada NSERC. In this section, all spaces are topological spaces which are path-connected, locally path-connected, and semilocally simply-connected. These are the the standard hypothesesfor the existence of universal covers. All maps between spaces are continuous. Forstandard results on covering space theory we refer the reader to Hatcher’s textbook onAlgebraic Topology [10].
Definition 2.1
Let X be a G –space and let Y be an H –space. A map f : X → Y is equivariant with respect to a group homomorphism f : G → H if f ( g . x ) = f ( g ) f ( x )for every g ∈ G and x ∈ X .If Y is an H –space then the universal cover e Y is naturally an e H –space and the coveringmap e Y → Y is equivariant with respect to a natural group homomorphism e H → H as the theorem below states. The proof is patterned after an argument by Bridson andHaefliger in the context of complexes of groups [3, Chap. III.C 1.15]. Theorem 2.2 (Lifting an Action to the Universal Cover) Let Y bean H –space, andlet ρ : e Y → Y be the universal covering map. Then there is a group e H and action e H × e Y → e Y withthe following properties.(1) There is anexact sequence of groups1 −→ π Y −→ e H −→ H −→ . (2) The covering map e Y → Y is equivariant withrespect to e H → H .(3) The restriction of e H × e Y → e Y to π Y × e Y → e Y is the standard action by decktransformations of π Y on e Y .(4) For each ˜ y ∈ ˜ Y mapping to y ∈ Y , the homorphism e H → H restricts to anisomorphism e H ˜ y → H y between the e H –stabilizer of ˜ y and the H –stabilizer of y .(5) If G × e Y → e Y is an action satisfying the four analogous properties above, thenthere isan isomorphism Φ : G → e H such that g . ˜ y = Φ ( g ) . ˜ y for all ˜ y ∈ e Y . A lgebraic & G eometric T opology XX (20XX) R.G. Hanlon and E. Mart´ınez-Pedroza (6) If f : X → Y isequivariant withrespectto f : G → H ,and X isone-connected,and ˜ f : X → e Y is a lifting of f , then ˜ f is equivariant with respect to a grouphomomorphism ˜ f : G → e H which lifts f : G → H .It is an immediate corollary that there exists liftings to intermediate covers which are H –regular as defined below. Definition 2.3 ( H –regular covers) Let Y be an H –complex. A covering map ˆ Y → Y is H –regular if the composition π b Y → π Y → e H is a normal subgroup of e H . Remark 2.4
Let Y be an H –complex. By definition, the universal cover of an H –complex is H –regular. Furthermore any cover of Y associated to a characteristicsubgroup of π ( Y ) is H –regular. Corollary 2.5 (Lifting an Action to an Intermediate Cover) Let Y be an H –spaceandlet b Y → Y bean H –regularcover. Thenthequotientgroup b H = e H /π b Y actson b Y and the map b Y → Y is equivariant with respect to b H −→ H . Furthermore, stabilizersof points are preserved in the sense that if ˆ y ∈ b Y maps to y ∈ Y then the restriction b H ˆ y → H y is anisomorphism. Proof of of Theorem 2.2
Let y be a point of Y , and recall that e Y can be identifiedwith the set e Y = { [ c ] | c is a path in Y starting at y } , where [ c ] denotes the homotopy class of c with respect to homotopies fixing theendpoints c (0) and c (1). The covering map e Y → Y is interpreted as sending [ c ] to c (1). The action of π ( Y , y ) on e Y is given by π ( Y , y ) × e Y −→ e Y , [ γ ] × [ c ] [ γ ∗ c ] . For details of this standard construction of the universal cover and the action of thefundamental group, we refer the reader to [10]. Let e H = { ( h , [ c ]) : h ∈ H , c a path in Y from y to h . y } . The group operation on e H is given by e H × e H −→ e H , ( h , [ c ]) × ( h ′ , [ c ′ ]) ( hh ′ , [ c ∗ h . c ′ ]) , where as usual ∗ denotes concatenation of paths. Observe that the operation is welldefined since for any pair of paths f ∈ [ c ] and f ′ ∈ [ c ′ ], the terminal point of f equalsthe initial point of hf ′ , and c ∗ h . c ′ is homotopic relative to endpoints to f ∗ h . f ′ , and A lgebraic & G eometric T opology XX (20XX)quivariant Towers and Non-positively Curved Groups the terminal point of c ∗ h . c ′ is hh ′ . y . To show that this operation endows e H with agroup structure is routine and it is left to the interested reader. The action of e H on e Y is given by e H × e Y −→ e Y , ( h , [ c ]) . [ c ′ ] = [ c ∗ h . c ′ ] , and one easily verifies that it is a well-defined action. Now we verify the six properties. Properties (1) , (2) and (3) Observe that the natural projection e H −→ H , ( h , [ c ]) h is a surjective homomorphism with kernel { (1 , [ c ]) | c is a closed path with c (0) = c (1) = y } ∼ = π ( Y , y ) . By definition the action e H × e Y → e Y extends the action of π ( Y , y ) on e Y . To verifythat e Y → Y is equivariant with respect to e H → H , let ( h ′ , [ c ′ ]) ∈ e H and [ c ] ∈ e Y andobserve that the terminal point of the path ( h ′ , [ c ′ ]) . [ c ] equals the point h ′ . c (1). Property (4) This statement follows from properties (1), (2) and (3) as follows. Let˜ y ∈ ˜ Y and let y ∈ Y be the image of ˜ y by the covering map. By equivariance, thehomomorphism e H → H maps the stabilizer e H ˜ y into H y . Let h ∈ H y and let e h ∈ ˜ H bean element mapping to h . Then there is g ∈ π Y such that g . ˜ y = ˜ h . ˜ y . It follows that g − ˜ h ∈ e H ˜ y and g − ˜ h maps to h . This shows that e H → H is surjective. For injectivity,if ˜ h , ˜ h ∈ ˜ H ˜ y map to h ∈ H y , then ˜ h − ˜ h ∈ π Y ∩ e H ˜ y . Since π Y acts freely on Y , itfollows that ˜ h = ˜ h . Property (5) The isomorphism is a consequence of the short five lemma. Supposethat G is another group acting on e Y and satisfying properties one to three; the fourthproperty is not needed as it is a consequence of the other three. By property (1) of G ,there is a short exact sequence1 −→ K −→ G ϕ −→ H −→ , where K ∼ = π ( Y ). Denote by ρ the covering map e Y → Y , and let ˜ y ∈ e Y be suchthat ρ (˜ y ) = y . For g ∈ G , let c g denote a path in Y starting at y obtained bycomposing ρ with a path in e Y from ˜ y to g . ˜ y ; here we use that G acts on e Y . Since e Y is simply-connected, the homotopy class [ c g ] depends only on g ; by property (2) of G , the pair ( ϕ ( g ) , [ c g ]) is an element of e H . It follows that there is a well defined grouphomomorphism Φ : G −→ e H , g ( ϕ ( g ) , [ c g ]) . A lgebraic & G eometric T opology XX (20XX) R.G. Hanlon and E. Mart´ınez-Pedroza
Observe that Φ ( K ) is a subset of the kernel of e H → H and therefore Φ satisfies thefollowing commutative diagram1 / / K / / Φ (cid:15) (cid:15) G ϕ / / Φ (cid:15) (cid:15) H id (cid:15) (cid:15) / / / / π ( Y , y ) / / e H / / H / / . We claim that the Φ : K → π ( Y , y ) is an isomorphism. By property (3) of G , theaction of K on e Y is the action of deck transformations of e Y → Y . Therefore for[ c ] ∈ π ( Y , y ), there is k ∈ K acting on e Y as (1 , [ c ]) does. Recall that [ c ] is identifiedwith (1 , [ c ]) ∈ e H . By definition of Φ , we have that Φ ( k ) = ( ϕ ( k ) , [ c ]) = (1 , [ c ]) andhence Φ : K → π ( Y , y ) is surjective. For injectivity, observe that if k ∈ K and Φ ( k )is trivial, then k ∈ G ˜ y ; then, by property (3) of G , k is trivial. By the short five lemmaapplied to the commutative diagram above, Φ : G → e H is an isomorphism.Now we verify that the actions of e H and G are identical up to composing with Φ ,i.e, for every g ∈ G and ˜ y ∈ e Y , g . ˜ y = Φ ( g ) . ˜ y . Suppose that ˜ y = [ c ′ ]. Then g . ˜ y = [ c g ∗ ϕ ( g ) . c ′ ] since c g ∗ ϕ ( g ) . c ′ lifts to a path from ˜ y to g . ˜ y . Hence g . ˜ y = [ c g ∗ ϕ ( g ) . c ′ ] = ( ϕ ( g ) , [ c g ]) . [ c ′ ]. Property (6) To simplify notation, we denote by f the map X → Y and the grouphomomorphism G → H . Let x ∈ X be such that f ( x ) = y . Since X is simplyconnected, using the description of e Y as a set of equivalence classes of paths, for any x ∈ X we have that ˜ f ( x ) = [ f ◦ c ] where c is any path in X from x to x . We showthat ˜ f : X → e Y is equivariant with respect to the group homomorphism˜ f : G → e H , g ( f ( g ) , [ f ◦ c ]) . where c is a path in X from x to g . x . Observe that ˜ f is well defined as a map since X is simply connected and f : X → Y is an equivariant map. To show that ˜ f is ahomomorphism is routine. To verify equivariance, first let x ∈ X and g ∈ G . Let c bea path from x to g . x , let c ′ be a path from x to g . x , and let c ′′ be a path from x to x . Since X is simply connected, we have that [ c ′ ∗ g . c ′′ ] = [ c ] and hence˜ f ( g ) . ˜ f ( x ) = ( f ( g ) , [ f ◦ c ′ ]) . [ f ◦ c ′′ ] = [( f ◦ c ′ ) ∗ ( f ( g ) . ( f ◦ c ′′ ))] = [ f ( c ′ ∗ g . c ′′ )] = [ f ( c )] = ˜ f ( g . x ) . This completes the proof of the theorem. A lgebraic & G eometric T opology XX (20XX)quivariant Towers and Non-positively Curved Groups For the rest of the paper, all spaces are combinatorial complexes and all maps are com-binatorial. All group actions on complexes are by combinatorial maps. A G –complex X is proper (respectively cocompact, free ) if the G -action is proper (respectively, co-compact, free). For a cell σ of X , the pointwise G –stabilizer of σ is denoted by G σ ,and the G -orbit of σ is denoted by G ( σ ). Definition 3.1 (Combinatorial Complexes and Maps) [3, Ch.I Appendix] A map X → Y between CW-complexes is combinatorial if its restriction to each open cell of X is a homeomorphism onto an open cell of Y . A CW-complex X is combinatorialprovided that the attaching map of each open cell is combinatorial for a suitablesubdivision. Definition 3.2 (Equivariant Map) Let X be a G –complex and let Y be an H –complex. An equivariant map f : X → Y is a pair ( f , f ) where f : X → Y is acombinatorial map, f : G → H is a group homomorphism, and f is equivariant withrespect to f , that is, f ( g . x ) = f ( g ) f ( x ) for every g ∈ G and x ∈ X . As usual, if both X and Y are G -complexes and f is the identity map on G , then f is called a G–map . Asa convention, we use bold letters to denote equivariant maps f , and in this case, we use f to denote the map between complexes, and f to denote the group homomorphism.The domain of an equivariant map is the space together with the group acting on it, andthe same convention applies to the codomain. The composition of equivariant maps isdefined in the natural way. Equality of equivariant maps f = g means their domainsand codomains are equal, f = g , and f = g . Definition 3.3 (Equivariant Isomorphism) An equivariant map is an isomorphism ifthe map at the level of spaces is a homeomorphism and the map a the level of groupsis a group isomorphism.
Definition 3.4 (Equivariant Inclusions) An equivariant map ı = ( ı, ı ) is called an equivariant inclusion if ı and ı are injective. The equivariant inclusion ı is called proper if either ı or ı is not surjective. Definition 3.5 (Equivariant Cover) Let Y be an H –complex. A covering map b Y → Y is called an equivariant cover if b Y is an H –regular cover. By Corollary 2.5, if ρ is an A lgebraic & G eometric T opology XX (20XX) R.G. Hanlon and E. Mart´ınez-Pedroza equivariant cover then there is a well defined group homomorphism ρ : b H → H suchthat ( ρ, ρ ) is an equivariant map. When refering to an equivariant cover we will beimplicitly referring to the associated equivariant map ( ρ, ρ ). Definition 3.6 (Towers and F –Towers) Let X be a G –complex and let Y be an H –complex. An equivariant map g : X → Y is an equivariant tower if it can be ex-pressed as an alternating composition of equivariant inclusions and equivariant covers.Specifically, g is a composition X = X n ֒ → b X n → X n − ֒ → · · · ֒ → b X → X ֒ → b X → X = Y and g is a composition G = G n ֒ → b G n → G n − ֒ → · · · ֒ → b G → G ֒ → b G → G = H where X i is a G i -complex, and b X i → X i − is a G i − -regular cover of X i − inducing the b G i -action on b X i , and X i is a subcomplex of b X i invariant under the subgroup G i ≤ b G i ,and both G = G n , and H = G . In the case that each X i is a full-subcomplex of b X i ,the tower g is called an F –tower .The length of the tower g is the smallest value of n in an expression for g as above. Inparticular, an equivariant inclusion or cover have length at most one. By convention,the identity map has length zero. Definition 3.7 (Equivariant Tower Lifting and F –tower Lifting) Let f be an equiv-ariant map. An equivariant tower lifting of f is an equivariant map f ′ such that there isan equivariant tower g such that f = g ◦ f ′ . The lifting is trivial if g is an equivariantisomorphism, and the lifting is maximal if the only equivariant tower lifting of f ′ is thetrivial one. The notions of equivariant F –tower lifting , trivial F –tower lifting , and maximal F –tower lifting are defined analogously. Remark 3.8 (Composition of Towers) Observethatif f and h areequivarianttowers,andthecodomainof f equalsthedomainof h (thismeansonthespaceandthegroup),then the composition h ◦ f is an equivariant tower. The same statement holds for F –towers.Suppose f = g ◦ f ′ isatowerliftingof f ,and f ′ = g ′ ◦ f ′′ isatowerliftingof f ′ . Sincethecomposition g ◦ g ′ isatower, f = ( g ◦ g ′ ) ◦ f ′′ isatowerlifting of f . Inparticular,if f ′′ isamaximaltowerlifting of f ′ then f ′′ isamaximaltowerliftingof f . Thesamestatement holds for F –towers liftings. Definition 3.9 (0–surjective) A map X → Y is 0 –surjective if every 0–cell of Y isin the image of X . A lgebraic & G eometric T opology XX (20XX)quivariant Towers and Non-positively Curved Groups Proposition 3.10 (Maximality ⇔ Surjectivity) Let X be a one-connected and G –cocompact complex, let f : X → Y bea G –map.(1) Anequivariant towerlifting f ′ of f ismaximalifandonlyif f ′ issurjective and π –surjective, and f ′ issurjective.(2) Anequivariant F –towerlifting f ′ of f ismaximalifandonlyif f ′ is 0–surjectiveand π –surjective, and f ′ is surjective. Proof
We sketch the proof for F –towers and the proof of the first statement is leftto the reader. The only if part is immediate since otherwise f ′ would factor through anon-trivial inclusion or the universal covering map. For the if part, let f ′ = h ◦ f ′′ be an F –tower lifting of f ′ . Suppose that h is an equivariant inclusion of a full-subcomplex;since f ′ is 0–surjective and f ′ is surjective we have that h and h are surjective andhence h is an equivariant isomorphism. Suppose that h is an equivariant cover; since X is one-connected and f ′ : X → Y ′ is π –surjective it follows that Y ′ is one-connectedand hence h is the trivial cover and h is an isomorphism. The general case followsby induction on the length of the tower. Definition 3.11 (Preserving 0–Stabilizers) An equivariant map f from the G –complex X to the H –complex Y is said to preserve –stabilizers if for every 0–cell σ of X the map f : G σ → H f ( σ ) is a group isomorphism. Definition 3.12 ( d ( f ) , r ( f ) , e ( X )) For a G –complex X , let v ( G , X ) denote the numberof G -orbits of 0–cells and let e ( G , X ) denote the number of G -orbits of 1–cells. Ifthe group is understood, we simply write v ( X ) and e ( X ). Observe that v ( G , X ) < ∞ and e ( G , X ) < ∞ if X is G –cocompact. If f : X → Y is an equivariant map fromthe G –complex X to the f ( G )-complex Y , we define d ( f ) = v ( G , X ) and r ( f ) = v ( f ( G ) , f ( X )). Observe that for a G –map f : X → Y we have that d ( f ) ≥ r ( f ) and e ( X ) ≥ e ( f ( X )). Lemma 3.13
Let g : X → Y beanequivariant towersuchthat g isanisomorphism,preserves 0–stabilizers, and ∞ > d ( g ) = r ( g ). Thefollowing statements hold.(1) Themap g is anequivariant inclusion andwhen restricted to the 0–skeletons isan isomorphism.(2) If g is surjective, then g isanequivariant isomorphism.(3) If ∞ > e ( X ) = e ( Y ), then g is an isomorphism when restricted to 1–skeletonsand, in particular, g is π –surjective. A lgebraic & G eometric T opology XX (20XX) R.G. Hanlon and E. Mart´ınez-Pedroza
Proof
Since g is an isomorphism, assume that X and Y are G -complexes and g : G → G is the identity map. Observe that the second statement is immediate if thefirst statement holds.For the first statement, we show first that g induces a bijection between the 0–skeletons.Since ∞ > d ( g ) = r ( g ), the pigeon-hole argument shows that g induces a bijectionbetween G -orbits of 0–cells. Hence it is enough to show that for any 0–cell x inthe domain of g , the induced map between orbits g : G ( x ) → G ( g ( x )) is a bijection.By equivariance, g : G ( x ) → G ( g ( x )) is surjective. For injectivity, suppose there are x , x ′ ∈ X and h ∈ G such that h . x = x ′ and g ( x ) = g ( x ′ ). By equivariance, h is in the G –stabilizer of g ( x ). Since g : G x → G g ( x ) is the identity map, it follows that h ∈ G x and hence x ′ = x . We have proved that g induces a bijection between the 0–skeletons.Now we show that g is an inclusion of complexes. Let σ and σ ′ be two k -cells of X mapping to the same k -cell of Y . Since g is bijective on 0–cells, the cells σ and σ ′ have a common 0–cell in their closure. Since g is a tower, it is a locally injective map.Therefore σ and σ ′ are the same k -cell.The third statement is proved as follows. The first statement of the lemma impliesthat g induces an equivariant inclusion between the 1–skeletons, and an isomorphismbetween 0–skeletons. It remains to prove that the induced map between 1–skeletons issurjective. Since ∞ > e ( X ) = e ( Y ), the pigeon-hole argument shows that g induces abijection between G -orbits of 1–cells. For any 1–cell σ , equivariance implies that theinduced map between orbits g : G ( σ ) → G ( g σ ) is surjective. Therefore g is surjectiveon 1–cells, and hence the induced map between the 1–skeletons is an isomorphism. F –tower Liftings Theorem 3.14 (Maximal Equivariant F –Towers) Let f : X → Y be a G –map. If X is one-connected and G –cocompact and Y is locally finite, then f has a maximalequivariant F –tower lifting.Before the proof of the theorem we need a definition and a remark. Definition 3.15 (Span) The span of a subcomplex K ⊂ X , denoted by Span X ( K ), isthe smallest full subcomplex of X containing K . Remark 3.16 If Y is a G –complex and K ⊂ Y is a G –subcomplex, then Span X ( K )is a G –subcomplex of Y . If, in addition, Y is locally finite and K is G –cocompactthen Span X ( K ) is G –cocompact and in particular e ( Span X ( K )) isfinite. A lgebraic & G eometric T opology XX (20XX)quivariant Towers and Non-positively Curved Groups Y n + g n + (cid:31) (cid:127) / / e Y n / / Y n (cid:31) (cid:127) / / · · · / / Y g (cid:31) (cid:31) (cid:31) (cid:127) / / e Y / / Y (cid:31) (cid:127) / / g (cid:31) (cid:31) e Y / / Y (cid:31) (cid:127) g / / YX f C C ✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞ f I I ✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒ f U U ✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱ f a a ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ f n g g ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ f n + i i ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ Figure 1: The tower construction in the proof of Theorem 3.14
Proof of Theorem 3.14
Let f denote the G –map f . Let Y be the G –subcomplex Span ( f ( X )) of Y , let f : X → Y by the G –map defined as f ( x ) = f ( x ) for x ∈ X ,and let g : Y → Y be the G –equivariant inclusion. Since X is G –cocompact, ∞ > d ( f ) − r ( f ). Since Y is locally finite and X is G –cocompact, Y is locally finiteand ∞ > e ( Y ).For n ≥
1, suppose that for we have defined an F –tower lifting f n : X → Y n of f n − and an F –tower g n : Y n → Y n − such that g n ◦ f n = f n − and Y n is locally finiteand ∞ > d ( f n ) − r ( f n ) and ∞ > e ( Y n ). Let ρ n : e Y n → Y n be the universal coveringmap and let ˜f n : X → e Y n be the equivariant lifting of f n to the universal cover e Y n . ByTheorem 2.2, ˜f n is naturally a G –map. Let Y n + be the G –subcomplex Span (˜ f n ( X )) of e Y n and let f n + : X → Y n + the G –map defined by f n + ( x ) = ˜f n ( x ) for each x ∈ X . Let g n + : Y n + → Y n be the restriction of ρ n to Y n + . By construction, f n + : X → Y n + is 0–surjective. Since e Y n is locally finite and X is G –cocompact, Y n + is also locallyfinite and ∞ > e ( Y n + ). The construction is illustrated in Figure 1.Consider the well-ordered set consisting of pairs of positive integers N × N with thedictionary order ≤ dic . For each f n we assign an element | f n | of N × N defined as | f n | = (cid:16) d ( f ) − r ( f n ) , e ( Y n ) (cid:17) . Lemma 3.17 (Decreasing complexity) Thefollowing statements hold.(1) If Y n is simply-connected then f n : X → Y n is a maximal F –tower lifting of f : X → Y . (2) If | f n + | = dic | f n | then Y n issimply-connected. A lgebraic & G eometric T opology XX (20XX) R.G. Hanlon and E. Mart´ınez-Pedroza (3) Forevery n , | f n + | ≤ dic | f n | . Proof If Y n is simply-connected then f n is π –surjective. Since f n is also 0–surjective,Proposition 3.10 implies that f n is a maximal F –tower lifting of f . This proves thefirst statement.For the second statement, suppose that | f n + | = dic | f n | . We will show that g n + : Y n + → Y n is π –surjective using Lemma 3.13(3); then the proof concludes by observing that g n + factors through the simply-connected space e Y n and hence Y n is simply-connected.It remains to verify that g n + satisfies the hypotheses of Lemma 3.13(3). By con-struction, g n + : Y n + → Y n is a G –map and a tower; the assumption implies that ∞ > d ( g n + ) = r ( f n + ) = r ( f n ) = r ( g n + ) and ∞ > e ( Y n + ) = e ( Y n ); an inductionargument together with Theorem 2.2(4) shows that g n + preserves 0–stabilizers for n ≥ ∞ > d ( f ) = d ( f n + ) ≥ r ( f n + ) = d ( g n + ) ≥ r ( g n + ) ≥ r ( f n ) ≥ , and hence d ( f ) − r ( f n ) ≥ d ( f ) − r ( f n + ) ≥ . If r ( f n ) = r ( f n + ) then d ( g n + ) = r ( g n + ). Then Lemma 3.13(1) implies that g n + : Y n + → Y n is a G –equivariant inclusion and therefore e ( Y n ) ≥ e ( Y n + ). Itfollows that | f n + | ≤ dic | f n | .To conclude the proof of Theorem 3.14 observe that if for every n the complex Y n isnot simply-connected, then there is an infinite strictly decreasing sequence of elementsof ( N × N , ≤ dic ). Since this is impossible, there is n such that Y n is simply-connectedand then f n is a maximal F –tower lifting of f . Theorem 3.18 (Maximal Equivariant Towers) Let f : X → Y be a G –map. If X isone-connected and G –cocompact, then f has amaximal equivariant towerlifting. Sketch of the proof
The proof of Theorem 3.18 is the same as the proof of Theo-rem 3.14 with some simplifications. Define inductively the sequence of tower lift-ings f n + : X → Y n + where Y n is defined as the G –subcomplex ˜f n ( X ) of e Y n and A lgebraic & G eometric T opology XX (20XX)quivariant Towers and Non-positively Curved Groups Y = f ( X ). Since X is G –cocompact and f n is surjective, Y n is G –cocompact andhence d ( f n ) − r ( f n ) < ∞ . Then, the same argument, shows that d ( f n ) − r ( f n ) ≥ d ( f n + ) − r ( f n + ) . One verifies that if Y n is not simply-connected then the inequality above is strict.Specifically, if r ( f n ) = r ( f n + ) then g n + is a surjective G –equivariant tower preserving0–stabilizers and such that d ( g n + ) = r ( g n + ); then Lemma 3.13(2) implies that g n + is an isomorphism factoring through e Y n and hence Y n is simply-connected. If each Y n is not simply-connected then d ( f n ) − r ( f n ) defines a strictly decreasing infinite sequenceof natural numbers which is impossible. Therefore some Y n is simply-connected andthe corresponding f n is the desired maximal tower lifting of f . Theorem 4.1 If Y is a one-connected, proper and locally finite H –complex and G ≤ H is finitely presented, then there is a one-connected cocompact G –complex X and a G –equivariant F –tower X → Y .The proof of the theorem requires two lemmas. Lemma 4.2
Let G beafinitegraphofgroupssuchthatvertexgroupsarefinitelygener-atedandedgegroupsarefinite. If ϕ : π ( G ) → G isasurjectivegrouphomomorphisminto afinitely presented group, then Kernel ( ϕ ) isnormally finitely generated. Proof
The hypotheses on G imply that π ( G ) is finitely generated. Therefore thereis a surjective homomorphism F ψ → π ( G ) where F is a finite rank free group. Since G is finitely presented and F has finite rank, the kernel of ϕ ◦ ψ is normally finitelygenerated, say Kernel ( ϕ ◦ ψ ) = hh r , . . . , r m ii . Since Kernel ( ϕ ) = ψ ( Kernel ( ϕ ◦ ψ )),we have that Kernel ( ϕ ) = hh ψ ( r ) , . . . , ψ ( r m ) ii .Recall that a group is slender if all its subgroups are finitely generated, and a G –complex is almost proper if G acts properly on the complement of the 0–skeleton. Lemma 4.3 (One-connected Complex for Finitely Presented Subgroup) Let Y be aone-connected, almostproper H –complexsuchthat H –stabilizers ofcellsareslender.Suppose that G ≤ H is finitely presented. Then there exists a cocompact and one-connected G-2 –complex X , and an equivariant map f : X → Y such that f is theinclusion G ֒ → H and f is injective when restricted to 0–cell stabilizers. A lgebraic & G eometric T opology XX (20XX) R.G. Hanlon and E. Mart´ınez-Pedroza
Proof
First we construct the 1–skeleton X (1) of X as a G –equivariant cocompactsubcomplex of Y . Let { g i } mi = be a finite generating set for G and let y be a 0–cellof Y . Since Y is connected, for each 1 ≤ i ≤ m there is a combinatorial path γ i from y to g i y . Let D be the finite subcomplex D = γ ∪ · · · ∪ γ m of Y and let X (1) bethe union of all G –translates of D in Y . Then X (1) is a G –equivariant 1-dimensionalcocompact connected subcomplex of Y . Connectedness follows from the assumptionthat { g i } mi = generates G , and cocompactness from D being a finite subcomplex.Next we show that π X (1) is a normally finitely generated group. Invoking Theorem 2.2,consider the group e G acting on the universal cover T of X (1) such that T → X (1) isequivariant with respect to e G → G and π X (1) is isomorphic to the kernel of e G → G .By considering the barycentric subdivision T ′ of T , we have a cocompact actionwithout inversions of e G on the tree T ′ ; here cocompactness follows from X (1) being G –cocompact. By Theorem 2.2 (4) the group homomorphism g : e G x → G g ( x ) isinjective for every x ∈ T . Therefore 1–cell e G –stabilizers of T ′ are finite, since Y isan almost proper G –complex. Moreover, 0–cell e G –stabilizers of T ′ are isomorphicto either a subgroup of the G –stabilizer of a 0–cell of Y and hence finitely generatedby the slender hypothesis; or to a subgroup of the setwise e G –stabilizer of a 1–cell of Y and hence finite since Y is almost proper. Then the theory of Bass and Serre on actionson trees [21] implies that e G is isomorphic to the fundamental group of a finite graph ofgroups G with finite edge groups and finitely generated vertex groups. By Lemma 4.2,the kernel of e G → G is normally finitely generated.To conclude the proof, we paste finitely many G -orbits of 2-cells to X (1) to obtaina one-connected complex. Choose a 0–cell ˜ x of T as a basepoint. Since π X (1) is normally finitely generated, there is a finite collection { r i } qi = of based loops in X (1) such that π X (1) = hh r , . . . , r q ii . Since Y is simply-connected and X (1) is asubcomplex of Y , for each r i there is a disk-diagram D i → Y with boundary path r i .Let X be the complex obtained by attaching a copy of D i to X (1) along the closed path g . r i for each g ∈ G and each 1 ≤ i ≤ q . Observe that X is connected and simply-connected, and the G -action on X (1) naturally extends to a cocompact G -action on X .The equivariant inclusion X (1) ֒ → Y extends to a map X → Y equivariant with respectto G ֒ → H .The proof of the Theorem 4.1 is an application of Lemma 4.3 together with Theo-rem 3.14. Proof of Theorem 4.1
Observe that the hypotheses of Lemma 4.3 are satisfied since Y being proper implies that stabilizers of 0–cells are finite and proper implies almost A lgebraic & G eometric T opology XX (20XX)quivariant Towers and Non-positively Curved Groups proper. Therefore there is a one-connected cocompact G –complex X and a G –map f : X → Y . By Theorem 3.14, there is a maximal F –tower lifting f = g ◦ f ′ of f ,where g : X → Y . By Proposition 3.10 (2), f ′ is 0–surjective and π –surjective, and f ′ is surjective. It follows that X is one-connected and f ′ is an isomorphism, in particular f ′ is a G –map. Since Y is locally finite and g is a tower, it follows that X is locallyfinite. Since X is G –cocompact and f ′ : X → X is 0–surjective, we have that X is G –cocompact.Analogously to the argument in the proof of Theorem 4.1, one obtains the followingresult by combining Lemma 4.3 and Theorem 3.18. In this result, the complex Y isnot necessarily locally finite but the conclusion is weaker. Theorem 4.4
Let Y be a one-connected and almost proper H –complex such that H –stabilizers of cells are slender. If G ≤ H is finitely presented, then there is aone-connected cocompact G –complex X and a G –equivariant tower X → Y .A version of Theorem 4.4 appears in [18, Lem. 6.4] where is shown that a G –map withlocally finite target always factors as the composition of a surjective and π –surjective G –map followed by a G –equivariant immersion. Definition 5.1 (Near-immersion) A map X → Y is a near immersion if it is locallyinjective in the complement of the 0–skeleton of X . Definition 5.2 (Diagramatically Reducible Complex) [6] A 2–complex X is diagra-matically reducible if there are no near-immersions C → X , where C is a cell structurefor the 2–dimensional sphere.First we recall some properties of diagramatically reducible complexes in the proposi-tion below. Proposition 5.3 (1) Diagramatically reducible complexes areaspherical. A lgebraic & G eometric T opology XX (20XX) R.G. Hanlon and E. Mart´ınez-Pedroza (2) Covers and subcomplexes of diagramatically reducible complexes are diagra-matically reducible.(3) The barycentric subdivision of a diagramatically reducible complex is diagra-matically reducible.
Proof
The first statement is a result of Gersten [6, Rem. 3.2], the second statement istrivial since the composition of a near-immersion and an immersion is a near-immersion,and the third statement is due to Howie [6, Rem. 6.10].
Definition 5.4 (Free 1–cells and Collapsing) Let X be a 2–complex. A 1–cell e ofa subcomplex Z ≤ X is free if it belongs to the boundary of a 2–cell f of Z , and e does not belong to the boundary of a 2–cell = f in Z . In this case, collapsing Z alonge means to remove the interior of e and the interior of f .The following characterization of diagramatically reducible complex is a result of JonCorson. Theorem 5.5 (Characterization) [5, Thm. 2.1] A one-connected 2–complex isdiagramatically reducible if and only if every finite subcomplex is 1–dimensional orcontains afree 1–cell.
Remark 5.6 (Equivariant Collapsing and Inversions) Recallthatagroupactiononacomplex has no inversions if whenever a cell is fixed setwise by a group element thenit is fixed pointwise by the group element. Let Z be a G –complex without inversionsand suppose that e is a free 1–cell of Z that belongs to the boundary of the 2–cell f . Observe that for every g ∈ G the 1–cell g . e is free in Z . Since G acts withoutinversions, for every g ∈ G , the 2–cell g . f contains only one 1–cell in the G –orbitof e , namely, g . e . Therefore we can simultaneously collapse Z along g . e for every g ∈ G obtaining a G –equivariant subcomplex Z ′ of X .Corson also proves that if F is a finite group acting on a one-connected diagramaticallyreducible 2–complex X then the fixed point set X F of F is non-empty [5, Thm. 4.1].The following proposition shows that X F is also contractible. Proposition 5.7 (Contractible Fixed Point Sets) Let X be a one-connected diagra-matically reducible 2–complex. If F is a finite group acting on X without inversions,then the fixedpoint set X F of F isa non-empty contractible subcomplex. A lgebraic & G eometric T opology XX (20XX)quivariant Towers and Non-positively Curved Groups Proof
Since F acts without inversions, X F is a subcomplex of X . By Proposi-tion 5.3(1), it is enough to show that X F is one-connected. First we verify that X F is connected. Take two vertices x and x of X F . Since X is connected, there is anedge–path α in X between x and x . Let Y be the subcomplex of X defined as theunion of all the images of α under the action of F . Observe that Y is a connectedfinite 1-dimensional subcomplex of X invariant under the F -action. Construct a finiteone-connected F -complex Y as follows. Fix a basepoint of Y and let γ , . . . , γ n bea collection of closed paths in Y that generate π ( Y ). For each γ i , there is a diskdiagram D i → X with boundary path γ i → X . Let Y be the 2–complex obtainedby attaching a copy of D i to Y along k .γ i for each k ∈ F . Then Y is a finite one-connected 2–complex, the F -action on Y extends to an action on Y , and there is anatural F -equivariant map f : Y → X . By Theorem 3.18, there is a maximal equiv-ariant tower lifting Y f ′ → Z g → X of f . Since g is an immersion, Z is diagramaticallyreducible. Since F acts without inversions on X , it also acts without inversions on Z . By Theorem 5.5, it follows that if Z contains 2–cells then it has a free 1–cell e .After a finite number of F -equivariant collapses of Z one obtains a 1-dimensionalone-connected F -complex equivariantly immersed into X , see Remark 5.6. Withoutloss of generality, we can assume that Z is 1-dimensional. Then Z is a tree andtherefore F fixes pointwise an edge-path in Z between x and x . By equivariance ofthe map Y → X , F fixes pointwise a path between x and x .Now we verify that X F is simply-connected. Since X is simply-connected, consideran essential embedded closed path γ in X F with minimal area in X . Then there is adisk-diagram D → X with boundary γ of minimal area. Let g ∈ F with g = D ∪ gD → X is a near-immersion. Since this isimpossible, X F is simply connected. Remark 5.8 (Inversions and Connected Fixed Point Sets) Duringthereviewprocessof the article, the referee observed that in the proof of Proposition 5.7, one can provethat X F is connected without assuming that F acts without inversions. The sketch oftheargument isasfollows. Suppose that X F isnot connected and choose x and x atminimal distance in different connected components of X F . Consider a path γ in X between x and x . Since x and x are in different connected components, the path γ is not fixed by F and hence there is a non-trivial element g ∈ F that does not fix γ pointwise. For given γ and g ∈ F , there is a disk diagram D → X with boundarypath γ − g ( γ ). Among all these possible choices of γ , g and D , choose the ones thatminimize Area ( D ). It follows that the diagram D contains 2–cells and no cut-points.Let n be the order of the element g . Then one can glue together n copies of D , byidentifying g ( γ ) in the i -th copy of D with γ in the i + D , producing a A lgebraic & G eometric T opology XX (20XX) R.G. Hanlon and E. Mart´ınez-Pedroza sphere S . The map S → X which maps the i -th copy of D in S to g i ( D ) in X is anear-immersion by our minimality choices. This contradicts that X is diagramaticallyreducible. Definition 5.9
Let G be a group. A proper G –complex X is a model for EG if forevery finite subgroup of F ≤ G , the fixed point set X F is contractible. Theorem 5.10 (Diagramatically Reduced Groups) Let Y be a diagramatically re-ducible proper H-2 –complex. If G ≤ H is finitely presented then G admits a diagra-matically reduced 2–dimensional cocompact model for EH . Proof
By passing to a subdivision of Y we can assume that G acts without inversionswhile still assuming that Y is diagramatically reducible, see Proposition 5.3(3). ByTheorem 4.4, there is an equivariant immersion X → Y where X is a one-connectedcocompact G-2 –complex. Since Y is diagramatically reducible, Proposition 5.3(2)implies that X is diagramatically reducible as well. Since the H -action on Y is properand without inversions, the same properties hold for the G –action on X . Let K bea finite subgroup of G . Then Proposition 5.7 implies that the fixed point set X F iscontractible. -Complexes Definition 5.11 (Fine Graphs and Fine Complexes [1]) A 1-complex is fine if each1–cell is contained in only finitely many circuits of length n for each n . Equivalently,the number of embedded paths of length n between any pair of (distinct) 0–cells isfinite. A complex is fine if its 1–skeleton is fine. Definition 5.12 (Relatively Hyperbolic Groups [1]) A group G is hyperbolic relativeto a finite collection of subgroups P if G acts cocompactly, almost properly on aconnected, fine, δ –hyperbolic 1–complex, and P is a set of representatives of distinctconjugacy classes of vertex stabilizers such that each infinite stabilizer is represented. Proposition 5.13 (2–dimensional Relative Hyperbolicity) Let X beaone-connected,negatively curved,fine,cocompact, andalmostproper G-2 –complex. Then G isahy-perbolic group relative to a (hence any) collection of representatives of conjugacyclasses of 0–cell stabilizers. A lgebraic & G eometric T opology XX (20XX)quivariant Towers and Non-positively Curved Groups Proof
Since X is negatively curved and one-connected, it is a δ -hyperbolic space.Indeed, it is well known that a CAT ( κ )–space with κ < δ –hyperbolic, and in theconformal case X satisfies a linear isoperimetric inequality and hence the combinatorialmetric on its one skeleton is a δ –hyperbolic space [3, 7]. Since X is G –cocompact thereare finitely many types of 2–cells and hence the 1–skeleton X (1) is quasi-isometric to X ; in particular X (1) is a δ ′ -hyperbolic space. It follows that X (1) is endowed with a G –action satisfying the conditions of Definition 5.12. Proposition 5.14
Let Y be a diagramatically reducible one-connected cocompactalmostproper H-2 -complexwithfineone-skeleton. If X isone-connected andthereisanimmersion X → Y then X has fineone-skeleton. Proof
We use the following characterization of simplicial fine graphs due to BrianBowditch [1, Prop. 2.1]. By simplicial we mean no double edges and no single edgeloops. A simplicial graph K is fine if and only if for each vertex x ∈ K , the set V ( x )of vertices adjacent to x has the following property: every subset of V ( x ) which isbounded in K \ { x } with respect to the combinatorial metric is finite.Without loss of generality assume that the boundary path of every 2–cell of Y is anembedded path, and that X and Y have simplicial one-skeleton. Indeed, by consideringthe barycentric subdivisions of X and Y , we can assume the boundary paths of 2–cells are embedded and one-skeletons are simplicial. Proving the proposition for thebarycentric subdivisions is sufficient since the one-skeleton of an almost proper H-2 -complex is fine if and only if the one-skeleton of its barycentric subdivision is fine,this follows directly from [1, Lem. 2.3] or [17, Lem. 2.9]; moreover, a complex isdiagramatically reducible if and only if its barycentric subdivision is diagramaticallyreducible, see Proposition 5.3(3).Since Y admits a cocompact H –action and its one skeleton is fine, for every n ∈ N there are finitely many cycles of length n up to the H –action. Therefore, since Y issimply-connected, there is a well defined Dehn function ∆ : N → N , that is, ∆ ( n ) isan upper bound for the area of minimal area disk diagrams with given boundary pathof length ≤ n .Let f : X → Y be an immersion. Let x ∈ X be 0–cell of X and let A be a subsetof 0–cells adjacent to x . Denote by diam ( A ) the diameter of A in X \ { x } in thecombinatorial metric, and analogously let diam ( f ( A )) denote the diameter of f ( A ) in Y \ { f ( x ) } in the combinatorial metric. The claim is that if diam ( A ) is finite then(1) diam ( f ( A )) ≤ C · ∆ ( diam ( A ) + , A lgebraic & G eometric T opology XX (20XX) R.G. Hanlon and E. Mart´ınez-Pedroza where C is an upper bound for the boundary length of a 2–cell of Y ; here C isfinite since Y admits a cocompact action. Assuming the claim, we conclude usingBowditch’s characterization of fineness as follows. If diam ( A ) is finite, then the claimimplies that diam ( f ( A )) is finite; then Y being fine implies that f ( A ) is a finite set; since f is an immersion and one-skeletons are simplicial, the induced map f : A → f ( A ) is abijection and hence A is finite set.Suppose that diam ( A ) = m < ∞ in X \ { x } . Let a , b ∈ A and suppose that a = b .Then there is a combinatorial path γ in X \ { x } from a to b of length ≤ m . Ifthe path f ◦ γ in Y does not contain the 0–cell f ( x ) then the combinatorial distancebetween f ( a ) and f ( b ) is bounded by m ; however this assumption on f ◦ γ might nothold. A general argument is as follows. Consider the closed path γ ′ → X definedas the concatenation γ ′ = e ∗ γ ∗ e where e is an 1–cell from x to a , and e isa 1–cell from b to x . Since X is simply-connected, there is a near-immersion of adisk-diagram D → X with boundary path γ ′ . Observe that there is only one 0–cell in ∂ D mapping to x ∈ X ; by abuse of notation, let x denote this 0–cell of D .The main observation is that D \{ x } is connected. Indeed, if D \{ x } is not connected,then the boundary path ∂ D → X pass through x more than once. Since e → X and e → X are 1–cells with only one endpoint equal x , and the image of γ → X doesnot contain x , it follows that ∂ D = e ∗ γ ∗ e pass through x only once, and hence D \ { x } is connected.The fact that D \ { x } is connected implies that there is an embedded path η → D between the two 0–cells adjacent to x in ∂ D , this path η → D factors through D \ { x } and goes around the 2–cells of D adjacent to x ; see Figure 2. Observe that η → D → X f → Y is a path between f ( a ) and f ( b ), and the combinatorial length of η is bounded by C · Area ( D ) where C is the upper bound for the boundary length of a2–cell of Y .Now observe that the path η → D → X f → Y does not intersect f ( x ). Indeed, sinceboundary paths of 2–cells of Y (and hence of D ) are embedded, if η → Y intersects f ( x ) then there is a 2–cell R of D whose boundary path contains x ∈ ∂ D and another0–cell x ∈ η both mapping to f ( x ); this would imply that ∂ R → Y is not an embeddedpath which is impossible by our initial assumption.Since Y is diagramatically reducible and D → Y is a near-immersion, it follows that D → Y is a minimal area disk diagram for ∂ D → Y . Therefore | η | ≤ C · ∆ ( | ∂ D | ),and hence the combinatorial distance between f ( a ) and f ( b ) in Y \ { f ( x ) } is boundedby | η | ≤ C · ∆ ( diam ( A ) + a and b were arbitrary, we have proved thatinequality (1) holds. A lgebraic & G eometric T opology XX (20XX)quivariant Towers and Non-positively Curved Groups diagram100. { ps,eps } not found (or no BBox) Figure 2: In the disk diagram D , the space D \ { x } is not connected. If D is a disk diagram, x is a 0–cell on ∂ D adjacent to the 0–cells a , b ∈ ∂ D with a = b , and D \ { x } is connected,then there is an edge path η → D between a , b that factors through D \ { x } and goes aroundthe 2–cells of D adjacent to x . The length of η is bounded by Area ( D ) · C where C is anupper bound for the boundary length of 2–cells of D . Theorem 5.15 (Subgroups of 2-Dim. Rel. Hyp. Groups are Rel. Hyp) Let Y bea one-connected negatively curved, fine, almost proper and cocompact H-2 –complexsuch that H –stabilizers of cells are slender. If G ≤ H is finitely presented then G ishyperbolic relative toafinite collection of G –stabilizers of cells of Y . Proof of Theorem 5.15
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