Box Spaces Research Articles

A Bott periodicity theorem for ℓp-spaces and the coarse Novikov conjecture at infinity

We formulate and prove a Bott periodicity theorem for an ℓp-space (1≤p<∞). For a proper metric space X with bounded geometry, especially for a coarsely connected space, we introduce a version of K-homology at infinity and the Roe algebra at infinity and show that to prove the coarse Novikov conjecture, it suffices to prove the coarse assembly map at infinity is an injection. As a result, we show that the coarse Novikov conjecture holds for any metric space with bounded geometry which admits a fibred coarse embedding into an ℓp-space. These include all box spaces of a residually finite hyperbolic group, and a large class of warped cones of a compact metric space with an action by a hyperbolic group.

The Bounded Isomorphism Conjecture for Box Spaces of Residually Finite Groups

The Bounded Isomorphism Conjecture for Box Spaces of Residually Finite Groups

The coarse Baum–Connes conjecture for certain relative expanders

Let (1→Nm→Gm→Qm→1)m∈N be a sequence of extensions of finite groups such that their coarse disjoint unions have bounded geometry. In this paper, we show that if the coarse disjoint unions of (Nm)m∈N and (Qm)m∈N are coarsely embeddable into Hilbert space, then the coarse Baum–Connes conjecture holds for the coarse disjoint union of (Gm)m∈N.As an application, the coarse Baum–Connes conjecture holds for the relative expanders constructed by G. Arzhantseva and R. Tessera, and the special box spaces of free groups discovered by T. Delabie and A. Khukhro, which do not coarsely embed into Hilbert space, yet do not contain a weakly embedded expander. This enlarges the class of metric spaces known to satisfy the coarse Baum–Connes conjecture. In particular, it solves an open problem raised by G. Arzhantseva and R. Tessera on the coarse Baum–Connes conjecture for relative expanders.

On arithmetic properties of solvable Baumslag–Solitar groups

Abstract For 0 &lt; α ≤ 1 0&lt;\alpha\leq 1 , we say that a sequence ( X k ) k &gt; 0 (X_{k})_{k&gt;0} of 𝑑-regular connected graphs has property D α D_{\alpha} if there exists a constant C &gt; 0 C&gt;0 such that diam ⁡ ( X k ) ≥ C ⋅ | X k | α \operatorname{diam}(X_{k})\geq C\cdot\lvert X_{k}\rvert^{\alpha} . We investigate property D α D_{\alpha} for arithmetic box spaces of the solvable Baumslag–Solitar groups BS ⁡ ( 1 , m ) \operatorname{BS}(1,m) (with m ≥ 2 m\geq 2 ): these are box spaces obtained by embedding BS ⁡ ( 1 , m ) \operatorname{BS}(1,m) into the upper triangular matrices in GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) \mathrm{GL}_{2}(\mathbb{Z}[1/m]) and intersecting with a family M N k M_{N_{k}} of congruence subgroups of GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) \mathrm{GL}_{2}(\mathbb{Z}[1/m]) , where the levels N k N_{k} are coprime with 𝑚 and N k ∣ N k + 1 N_{k}\mid N_{k+1} . We prove that if an arithmetic box space has D α D_{\alpha} , then α ≤ 1 2 \alpha\leq\frac{1}{2} ; if the family ( N k ) k (N_{k})_{k} of levels is supported on finitely many primes, the corresponding arithmetic box space has D 1 / 2 D_{\smash{1/2}} ; if the family ( N k ) k (N_{k})_{k} of levels is supported on a family of primes with positive analytic primitive density, then the corresponding arithmetic box space does not have D α D_{\alpha} for every α &gt; 0 \alpha&gt;0 . Moreover, we prove that if we embed BS ⁡ ( 1 , m ) \operatorname{BS}(1,m) in the group of invertible upper-triangular matrices T n ⁢ ( Z ⁢ [ 1 / m ] ) T_{n}(\mathbb{Z}[1/m]) , then every finite index subgroup of the embedding contains a congruence subgroup. This is a version of the congruence subgroup property (CSP).

TA`TSIRU ISTIKHDAAMU WASAAITHA AL-GHAAZI AL-KALIMAAT AL-MUTAQAATHI’AH ILAA MAHAARAH KITAABAH AL-LUGHAH AL-‘ARABIYYAH LISHAF AT-TSAMIN FII AL-MADRASAH AT-TSANAWIYYAH “DARUL IHSAN” CIBUNGBULANG

Students often assume that learning Arabic is difficult. Game are an activity that has rules, purpose and fun. The playing of crossword puzzle is one of the Arabic learning media where students must fill the blank (white box) spaces with letters that from a word based on the instructions given. In addition, the game can be developed to train students to make perfect sentences from the root words that result from a crossword puzzle answer paragraph. This research is meant to know impact on use of the medium crossword puzzle on the Arabic writing skills of class eight Junior High School Darul Ihsan Cibungbulang. The approach used in the study is the quantitative with experimental methods, While the research design uses a Pre-Experimental type One-Group Pretest Posttest Degsain. Data sources obtained from primary and secondary data. Data collection techniques use observation, interview, angket and test. Angket results and test are then treated using SPSS (Stastical Program For Sosial Sciences) for windows versi 26 dan Microsoft Excel 2010. Hypotheses in the test using Paired Sample t-Test. Based on data analysis, this studysuggests that the use of a medium of crossword puzzle can affect the ability of the Arabic writing skills of class eight Junior High School Darul Ihsan Cibungbulang, with a Sig.(2-tailed) sum of 0,000&lt;0,05

Measured Asymptotic Expanders and Rigidity for Roe Algebras

Abstract In this paper, we give a new geometric condition in terms of measured asymptotic expanders to ensure rigidity of Roe algebras. Consequently, we obtain the rigidity for all bounded geometry spaces that coarsely embed into some $L^p$-space for $p\in [1,\infty )$. Moreover, we also verify rigidity for the box spaces constructed by Arzhantseva–Tessera and Delabie–Khukhro even though they do not coarsely embed into any $L^p$-space. The key step in our proof of rigidity is showing that a block-rank-one (ghost) projection on a sparse space $X$ belongs to the Roe algebra $C^{\ast }(X)$ if and only if $X$ consists of (ghostly) measured asymptotic expanders. As a by-product, we also deduce that ghostly measured asymptotic expanders are new sources of counterexamples to the coarse Baum–Connes conjecture.

Effects of inter-reflections in box spaces on perceived object color harmony and shape

Effects of inter-reflections in box spaces on perceived object color harmony and shape

English

We classify which dual functors on a unitary multitensor category are compatible with the dagger structure in terms of groupoid homomorphisms from the universal grading groupoid to $\mathbb{R}_{>0}$ where the latter is considered as a groupoid with one object. We then prove that all unitary dual functors induce unitarily equivalent bi-involutive structures. As an application, we provide the unitary version of the folklore correspondence between shaded planar ${\rm C^*}$ algebras with finite dimensional box spaces and unitary multitensor categories with a chosen unitary dual functor and chosen generator. We make connection with the recent work of Giorgetti-Longo to determine when the loop parameters in these planar algebras are scalars. Finally, we show that we can correct for many non-spherical choices of dual functor by adding the data of a spherical state on $\operatorname{End}_{\mathcal{C}}(1_{\mathcal{C}})$, similar to the spherical state for a graph planar algebra.

Combinatorial cost: A coarse setting

The main inspiration for this paper is a paper by Elek where he introduces combinatorial cost for graph sequences. We show that having cost equal to 1 1 and hyperfiniteness are coarse invariants. We also show that “cost − 1 -1 ” for box spaces behaves multiplicatively when taking subgroups. We show that graph sequences coming from Farber sequences of a group have property A if and only if the group is amenable. The same is true for hyperfiniteness. This generalises a theorem by Elek. Furthermore we optimise this result when Farber sequences are replaced by sofic approximations. In doing so we introduce a new concept: property almost-A.

Coarse fundamental groups and box spaces

AbstractWe use a coarse version of the fundamental group first introduced by Barcelo, Kramer, Laubenbacher and Weaver to show that box spaces of finitely presented groups detect the normal subgroups used to construct the box space, up to isomorphism. As a consequence, we have that two finitely presented groups admit coarsely equivalent box spaces if and only if they are commensurable via normal subgroups. We also provide an example of two filtrations (Ni) and (Mi) of a free group F such that Mi &gt; Ni for all i with [Mi:Ni] uniformly bounded, but with $\squ _{(N_i)}F$ not coarsely equivalent to $\squ _{(M_i)}F$. Finally, we give some applications of the main theorem for rank gradient and the first ℓ2 Betti number, and show that the main theorem can be used to construct infinitely many coarse equivalence classes of box spaces with various properties.

Sofic boundaries of groups and coarse geometry of sofic approximations

Sofic groups generalise both residually finite and amenable groups, and the concept is central to many important results and conjectures in measured group theory. We introduce a topological notion of a sofic boundary attached to a given sofic approximation of a finitely generated group and use it to prove that coarse properties of the approximation (property A, asymptotic coarse embeddability into Hilbert space, geometric property (T)) imply corresponding analytic properties of the group (amenability, a-T-menability and property (T)), thus generalising ideas and results present in the literature for residually finite groups and their box spaces. Moreover, we generalise coarse rigidity results for box spaces due to Kajal Das, proving that coarsely equivalent sofic approximations of two groups give rise to a uniform measure equivalence between those groups. Along the way, we bring to light a coarse geometric viewpoint on ultralimits of a sequence of finite graphs first exposed by Ján Špakula and Rufus Willett, as well as proving some bridging results concerning measure structures on topological groupoid Morita equivalences that will be of interest to groupoid specialists.

Warped cones, (non‐)rigidity, and piecewise properties

We prove that if a quasi-isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi-isometry of the respective warped cones. For a general quasi-isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi-isometric after taking Cartesian products with suitable powers of the integers. Secondly, we characterise geometric properties of a group (coarse embeddability into Banach spaces, asymptotic dimension, property A) by properties of the warped cone over an action of this group. These results apply to arbitrary asymptotically faithful coverings, in particular to box spaces. As an application, we calculate the asymptotic dimension of a warped cone, improve bounds by Szab\'o, Wu, and Zacharias and by Bartels on the amenability dimension of actions of virtually nilpotent groups, and give a partial answer to a question of Willett about dynamic asymptotic dimension. In the appendix, we justify optimality of the aforementioned result on general quasi-isometries by showing that quasi-isometric warped cones need not come from quasi-isometric groups, contrary to the case of box spaces.

Box spaces of the free group that neither contain expanders nor embed into a Hilbert space

We construct box spaces of a free group that do not coarsely embed into a Hilbert space, but do not contain coarsely nor weakly embedded expanders. We do this by considering two sequences of subgroups of the free group: one which gives rise to a box space which forms an expander, and another which gives rise to a box space that can be coarsely embedded into a Hilbert space. We then take certain intersections of these subgroups, and prove that the corresponding box space contains generalized expanders. We show that there are no weakly embedded expanders in the box space corresponding to our chosen sequence by proving that a box space that covers another box space of the same group that is coarsely embeddable into a Hilbert space cannot contain weakly embedded expanders.

From the geometry of box spaces to the geometry and measured couplings of groups

In this paper, we prove that if two “box spaces” of two residually finite groups are coarsely equivalent, then the two groups are “uniform measured equivalent” (UME). More generally, we prove that if there is a coarse embedding of one box space into another box space, then there exists a “uniform measured equivalent embedding” (UME-embedding) of the first group into the second one. This is a reinforcement of the easier fact that a coarse equivalence (resp.ã coarse embedding) between the box spaces gives rise to a coarse equivalence (resp.ã coarse embedding) between the groups. We deduce new invariants that distinguish box spaces up to coarse embedding and coarse equivalence. In particular, we obtain that the expanders coming from [Formula: see text] cannot be coarsely embedded inside the expanders of [Formula: see text], where [Formula: see text] and [Formula: see text]. Moreover, we obtain a countable class of residually finite groups which are mutually coarse-equivalent but any of their box spaces are not coarse-equivalent.

The asymptotic dimension of box spaces of virtually nilpotent groups

We show that every box space of a virtually nilpotent group has asymptotic dimension equal to the Hirsch length of that group.

Rokhlin dimension for actions of residually finite groups

We introduce the concept of Rokhlin dimension for actions of residually finite groups on$\text{C}^{\ast }$-algebras, extending previous such notions for actions of finite groups and the integers by Hirshberg, Winter and the third author. We are able to extend most of their results to a much larger class of groups: those admitting box spaces of finite asymptotic dimension. This latter condition is a refinement of finite asymptotic dimension and has not previously been considered. In a detailed study we show that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a large class of examples. We show that actions with finite Rokhlin dimension by groups with finite-dimensional box spaces preserve the property of having finite nuclear dimension when passing to the crossed product$\text{C}^{\ast }$-algebra. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of infinite, finitely generated, nilpotent groups on finite-dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group$\text{C}^{\ast }$-algebras have finite nuclear dimension. This extends an analogous result about$\mathbb{Z}^{m}$-actions by the first author to a significantly larger class of groups, showing that a large class of crossed products by actions of such groups fall under the remit of the Elliott classification programme. We also provide results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing$\text{C}^{\ast }$-algebra.

Full box spaces of free groups

Abstract In this paper we investigate full box spaces and coarse equivalences between them. We do this in two parts. In part one we compare the full box spaces of free groups on different numbers of generators. In particular, the full box space of a free group F k {F_{k}} is not coarsely equivalent to the full box space of a free group F d {F_{d}} , if d ≥ 8 ⁢ k + 10 {d\geq 8k+10} . In part two we compare □ f ⁢ ℤ n {\Box_{f}\mathbb{Z}^{n}} to the full box spaces of 2-generated groups. In particular, we prove that the full box space of ℤ n {\mathbb{Z}^{n}} is not coarsely equivalent to the full box space of any 2-generated group, if n ≥ 3 {n\geq 3} .

Expanders and box spaces

We consider box spaces of finitely generated, residually finite groups G, and try to distinguish them up to coarse equivalence. We show that, for n≥2, the group SLn(Z) has a continuum of box spaces which are pairwise non-coarsely equivalent expanders. Moreover, varying the integer n≥3, expanders given as box spaces of SLn(Z) are pairwise not coarsely equivalent; similarly, varying the prime p, expanders given as box spaces of SL2(Z[p]) are pairwise not coarsely equivalent. We also show that, for prime p, the family of finite groups (PSL2(Z/pnZ))n≥1 can be turned in infinitely many ways (up to coarse equivalence) into a 6-regular expander.A strong form of non-expansion for a box space is the existence of α∈]0,1] such that the diameter of each component Xn satisfies diam(Xn)=Ω(|Xn|α). By [2], the existence of such a box space implies that G virtually maps onto Z: we establish the converse. For the lamplighter group (Z/2Z)≀Z and for a semi-direct product Z2⋊Z, such box spaces are explicitly constructed using specific congruence subgroups.We finally introduce the full box space of G, i.e. the metric disjoint union of all finite quotients of G. We prove that the full box space of a group mapping onto the free group F2 is not coarsely equivalent to the full box space of an S-arithmetic group satisfying the Congruence Subgroup Property.

Fibred coarse embeddability of box spaces and proper isometric affine actions on $L^p$ spaces

We show the necessary part of the following theorem : a finitely generated, residually finite group has property $PL^p$ (i.e. it admits a proper isometric affine action on some $L^p$ space) if, and only if, one (or equivalently, all) of its box spaces admits a fibred coarse embedding into some $L^p$ space (sufficiency is due to [CWW13]). We also prove that coarse embeddability of a box space of a group into a $L^p$ space implies property $PL^p$ for this group.

L p -distortion and p -spectral gap of finite graphs

We give a lower bound for the L p -distortion c p ( X ) of finite graphs X, depending on the first eigenvalue λ 1 ( p ) ( X ) of the p-Laplacian and the maximal displacement of permutations of vertices. For a k-regular vertex-transitive graph it takes the form c p ( X ) p ⩾ diam ( X ) p λ 1 ( p ) ( X ) 2 p − 1 k . This bound is optimal for expander families and, for p = 2 , it gives the exact value for cycles and hypercubes. As new applications we give non-trivial lower bounds for the L 2 -distortion for families of Cayley graphs of the finite lamplighter groups C 2 ≀ C n d ( d ⩾ 2 fixed), and for a family of Cayley graphs of SL n ( q ) ( q fixed, n ⩾ 2 ) with respect to a standard two-element generating set. An application to the L 2 -compression of certain box spaces is also given.