Asymptotic property C is a dimension-like large-scale invariant of metric spaces that is of interest when applied to spaces with infinite asymptotic dimension. It was first described by Dranishnikov, who based it on Haver\u27s topological property C. Topological property C fails to be preserved by products in very striking ways and so a natural question that remained open for some 10+ years is whether asymptotic property C is preserved by products. Using a technique inspired by Rohm we show that asymptotic property C is preserved by direct products of metric spaces

Bell, Gregory C.

Nagórko, Andrzej

English

University of Dayton

University of DaytoneCommonsSummer Conference on Topology and ItsApplications Department of Mathematics6-2017On Product Stability of Asymptotic Property CGregory C. BellUniversity of North Carolina at Greensboro, gcbell@uncg.eduAndrzej NagórkoUniversity of WarsawFollow this and additional works at: http://ecommons.udayton.edu/topology_confPart of the Geometry and Topology Commons, and the Special Functions CommonsThis Topology + Geometry is brought to you for free and open access by the Department of Mathematics at eCommons. It has been accepted forinclusion in Summer Conference on Topology and Its Applications by an authorized administrator of eCommons. For more information, pleasecontact frice1@udayton.edu, mschlangen1@udayton.edu.eCommons CitationBell, Gregory C. and Nagórko, Andrzej, "On Product Stability of Asymptotic Property C" (2017). Summer Conference on Topology andIts Applications. 28.http://ecommons.udayton.edu/topology_conf/28Asymptotic Property COn Product Stability of Asymptotic Property CGreg Bell∗, Andrzej Nagórko32nd Summer Conference on Topology and Its ApplicationsJune 2017Asymptotic Property CPlan of the TalkMotivationLarge-Scale GeometryProduct StabilityDefinitionsAsymptotic DimensionAsymptotic Property CMain ResultRohm’s TechniqueFurther ResultsAsymptotic Property CMotivationLarge-Scale GeometryLarge-scale dimensionI Motivation comes from Gromov’s approach to groups as metricspaces.I Discrete metric spaces have uninteresting topology.I However, Gromov considered the topological properties of suchdiscrete spaces “when viewed from infinitely far away.”ExampleR and Z look the same when viewed from infinitely far away.I This gives rise to large-scale topological properties, likelarge-scale connectedness, or large-scale dimension.Asymptotic Property CMotivationProduct StabilityProduct StabilityI Given a large-scale topological property, it is natural to want toknow whether this property is preserved by products.I Thus, if G and H have propertyP . . .I Does G×H haveP?I Does G ∗H haveP?ExampleGiven two spaces with finite large-scale dimension, their product willhave finite large-scale dimension.I We make these notions more precise presently.Asymptotic Property CDefinitionsAsymptotic DimensionUniformly bounded and R-disjointDefinitionA family U of subsets of a metric space X is said to be uniformlybounded if sup{diam(U) | U ∈ U} <∞.DefinitionLet R > 0 be given. A family of subsets U of X is said to be R-disjointif d(U, V ) > R whenever U 6= V are elements of U . Here,d(U, V ) = inf{d(u, v) | u ∈ U, v ∈ V }.Asymptotic Property CDefinitionsAsymptotic DimensionAsymptotic DimensionDefinitionA metric space X is said to have asymptotic dimension no more thann, expressed asdimX ≤ n if ∀R > 0 ∃ n+ 1-many families ofsubsets of X U0,U1, . . . ,Un such that1. diam(U) is uniformly bounded for U ∈ Ui, (i = 0, 1, . . . , n);2. each family Ui is R-disjoint; and3. U0 ∪ · · · ∪ Un covers X .Asymptotic Property CDefinitionsAsymptotic DimensionasdimT = 1asdimF2 = 1.I Cover the Cayley graph ofF2 by concentric 3R-thickannuli alternating red andblue.I Then, take R-connectedcomponents.I We have 2 uniformlybounded R-disjoint familiesof subsets that cover.Asymptotic Property CDefinitionsAsymptotic DimensionGroups with (In)Finite Asymptotic DimensionExamples.The following is a (non-exhaustive) list of classes of groups with finiteasdim.I Hyperbolic GroupsI Braid GroupsI Arithmetic GroupsI Relatively hyperbolic...I Artin GroupsI Coxeter GroupsI Mapping Class GroupsI One-relator GroupsExamples.I Thompson’s group F has infinite asymptotic dimension.I The first Grigorchuk group has infinite asymptotic dimension.Asymptotic Property CDefinitionsAsymptotic Property CAsymptotic Property CDefinition (Dranishnikov)A metric space X has asymptotic property C if for any numbersequence R1, R2, . . . there is an integer n and families of subsetsU0,U1, . . . ,Un of X such that1. diam(U) is uniformly bounded for U ∈ Ui, (i = 0, 1, . . . , n); and2. each Ui is Ri-disjoint; and3. U0 ∪ · · · ∪ Un covers X .TheoremSpaces with finite asymptotic dimension have asymptotic property C.Asymptotic Property CDefinitionsAsymptotic Property CExamplesI⊕nZ has APC (Yamauchi).I⊔nZn can be metrized to have APC.I Are there finitely generated groups with infinite asymptoticdimension and APC?Asymptotic Property CMain ResultAsymptotic property C is preserved by direct productsTheorem (B.-Nagórko, Davilla)Let X and Y be metric spaces with asymptotic property C. Then,X × Y has asymptotic property C.Compare to Topological Property CI (E. Pol & R. Pol, 2009) There exists a separable complete metricC-space X such that X2 fails to have C.I (R. Pol, 1986) There is a separable metric C-space whoseproduct with the irrationals fails to have C.I (Rohm, 1990) If X and Y are compact C-spaces, then X × Yhas C.Asymptotic Property CMain ResultRohm’s TechniqueAsymptotic Dimension of ProductsThe simple way to prove that the product of spaces with finiteasymptotic dimension has finite asymptotic dimension is suggested bythe following picture.YXGiven a cover uniformlybounded R-disjoint cover ofX and a uniformly boundedR-disjoint cover for Y we canconstruct a uniformlybounded cover for X × Ywith cardinality(asdimX+1)(asdimY +1).Asymptotic Property CMain ResultRohm’s TechniqueAsymptotic Property CUnfortunately, such a picture won’t work to show APC:I the number of sets n for X and the number of sets m for Y maychange with the sequence R1, R2, . . . andI the disjointness of these rectangles is governed by the minimumof the disjointness in the X-factor and the Y -factor.Asymptotic Property CMain ResultRohm’s TechniqueSketch of the ProofLet R1 ≤ R2 ≤ · · · be a given sequence of positive numbers.Rearrange the sequence into subsequences Ri,1 ≤ Ri,2 ≤ · · · likethis:R1,1R1R1,2R3R1,3R6R1,4R10R1,5R15R1,6R21R2,1R2R2,2R5R2,3R9R2,4R14R2,5R20R2,6R27R3,1R4R3,2R8R3,3R13R3,4R19R3,5R26R3,6R34R4,1R7R4,2R12R4,3R18R4,4R25R4,5R33R4,6R42R5,1R11R5,2R17R5,3R24R5,4R32R5,5R41R5,6R51R6,1R16R6,2R23R6,3R31R6,4R40R6,5R50R6,6R61Asymptotic Property CMain ResultRohm’s TechniqueSketch of ProofR1,1R1,2R2,1R2,2R3,1R3,2R4,1R4,2R5,1R5,2R6,1R6,2R1,n1R2,n2R3,n3R4,n4R5,n5R6,n6{U1,j}n1j=1 {U2,j}n2j=1 {U3,j}n3j=1 {U4,j}n4j=1 {U5,j}n5j=1 {U6,j}n6j=1FindU i,jandniforRi,jFind Vi corresponding to Ri,niWe find covers {Ui,j}nij=1 of X for each column Ri,1, Ri,2, . . . , and then construct a cover{Vi}mi=1 of Y corresponding to Ri,niAsymptotic Property CMain ResultRohm’s TechniqueSketch of ProofI LetWi,j = {U × V : U ∈ Ui,j , V ∈ Vi}.I Check that the collection⋃i,jWi,j covers X × Y .I diam(W )2 ≤ mesh(Ui,j)2 +mesh(Vi)2.I Each familyWi,j is Ri,j-disjoint.I Finally, we rearrange theWi,j back into a single sequence asabove.Asymptotic Property CMain ResultFurther ResultsFibering TheoremOur fibering theorem was inspired by old results of Hurewicz on mapsthat lower dimension.Theorem (Hurewicz, 1927)If f : X → Y is a closed map of separable metric spaces and if thereis some k so that dim f−1(y)) ≤ k for all y ∈ Y , thendimX ≤ dimY + k.Asymptotic Property CMain ResultFurther ResultsFibering TheoremFibering resultsIn coarse geometry there are several results of the type:I Let f : X → Y be a coarse map.I Let Y have some coarse propertyP .I Assume that for all R > 0, f−1(BR(y)) has propertyPuniformly in y.Then X has property P .Theorem (B.-Nagórko)If f : X → Y is uniformly expansive, if Y has APC, and if coarsefibers of f have uniform APC, then X has APC.Asymptotic Property CMain ResultFurther ResultsFree productsAs a corollary of the fibering theorem we obtain:Theorem (B.-Nagórko)Let G and H be groups (in proper left-invariant metrics) with APC.Then G ∗H has APC.Using entirely different techniques, we can show this theorem for “freeproducts” of general spaces:Theorem (B.-Nagórko)Let X and Y be metric spaces with APC. Then X ∗ˆY has APC.

On Product Stability of Asymptotic Property C

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On Product Stability of Asymptotic Property C

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