Authors:Tüli̇n Altunöz, Mehmetci̇k Pamuk, Oguz Yildiz Pages: 1 - 24 Abstract: Journal of Topology and Analysis, Ahead of Print. For a nonorientable surface, the twist subgroup is an index [math] subgroup of the mapping class group generated by Dehn twists about two-sided simple closed curves. In this paper, we consider involution generators of the twist subgroup and give generating sets of involutions with smaller number of generators than the ones known in the literature using new techniques for finding involution generators. Citation: Journal of Topology and Analysis PubDate: 2020-10-19T07:00:00Z DOI: 10.1142/S1793525321500023
Authors:François Dahmani, Ruoyu Li Pages: 1 - 38 Abstract: Journal of Topology and Analysis, Ahead of Print. We prove that for a free product [math] with free factor system [math], any automorphism [math] preserving [math], atoroidal (in a sense relative to [math]) and none of whose power send two different conjugates of subgroups in [math] on conjugates of themselves by the same element, gives rise to a semidirect product [math] that is relatively hyperbolic with respect to suspensions of groups in [math]. We recover a theorem of Gautero–Lustig and Ghosh that, if [math] is a free group, [math] an automorphism of [math], and [math] is its family of polynomially growing subgroups, then the semidirect product by [math] is relatively hyperbolic with respect to the suspensions of these subgroups. We apply the first result to the conjugacy problem for certain automorphisms (atoroidal and toral) of free products of abelian groups. Citation: Journal of Topology and Analysis PubDate: 2020-10-06T07:00:00Z DOI: 10.1142/S1793525321500011
Authors:Chris Bourne, Alan L. Carey, Matthias Lesch, Adam Rennie Pages: 1 - 52 Abstract: Journal of Topology and Analysis, Ahead of Print. In this paper, we give a comprehensive treatment of a “Clifford module flow” along paths in the skew-adjoint Fredholm operators on a real Hilbert space that takes values in [math] via the Clifford index of Atiyah–Bott–Shapiro. We develop its properties for both bounded and unbounded skew-adjoint operators including an axiomatic characterization. Our constructions and approach are motivated by the principle that spectral flow = Fredholm index. That is, we show how the KO-valued spectral flow relates to a KO-valued index by proving a Robbin–Salamon type result. The Kasparov product is also used to establish a [math] result at the level of bivariant K-theory. We explain how our results incorporate previous applications of [math]-valued spectral flow in the study of topological phases of matter. Citation: Journal of Topology and Analysis PubDate: 2020-09-03T07:00:00Z DOI: 10.1142/S1793525320500557
Authors:Henry Adams, Mark Heim, Chris Peterson Pages: 1 - 27 Abstract: Journal of Topology and Analysis, Ahead of Print. Let [math] be a group acting properly and by isometries on a metric space [math]; it follows that the quotient or orbit space [math] is also a metric space. We study the Vietoris–Rips and Čech complexes of [math]. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes. Citation: Journal of Topology and Analysis PubDate: 2020-09-03T07:00:00Z DOI: 10.1142/S1793525320500569
Authors:Andreas Čap, Christoph Harrach, Pierre Julg Pages: 1 - 39 Abstract: Journal of Topology and Analysis, Ahead of Print. Let [math] be a semisimple Lie group with finite center, [math] a maximal compact subgroup, and [math] a parabolic subgroup. Following ideas of P. Y. Gaillard, one may use [math]-invariant differential forms on [math] to construct [math]-equivariant Poisson transforms mapping differential forms on [math] to differential forms on [math]. Such invariant forms can be constructed using finite-dimensional representation theory. In this general setting, we first prove that the transforms that always produce harmonic forms are exactly those that descend from the de Rham complex on [math] to the associated Bernstein–Gelfand–Gelfand (or BGG) complex in a well defined sense. The main part of this paper is devoted to an explicit construction of such transforms with additional favorable properties in the case that [math]. Thus, [math] is [math] with its natural CR structure and the relevant BGG complex is the Rumin complex, while [math] is complex hyperbolic space of complex dimension [math]. The construction is carried out both for complex and for real differential forms and the compatibility of the transforms with the natural operators that are available on their sources and targets are analyzed in detail. Citation: Journal of Topology and Analysis PubDate: 2020-08-28T07:00:00Z DOI: 10.1142/S1793525320500570
Authors:Yosuke Kubota Pages: 1 - 30 Abstract: Journal of Topology and Analysis, Ahead of Print. In this paper, we introduce the notion of almost flatness for (stably) relative bundles on a pair of topological spaces and investigate basic properties of it. First, we show that almost flatness of topological and smooth sense are equivalent. This provides a construction of an almost flat stably relative bundle on enlargeable manifolds. Second, we show the almost monodromy correspondence, that is, a correspondence between almost flat (stably) relative bundles and (stably) relative quasi-representations of the fundamental group. Citation: Journal of Topology and Analysis PubDate: 2020-08-18T07:00:00Z DOI: 10.1142/S1793525320500545
Authors:Gil Goffer, Gennady A. Noskov Pages: 1 - 22 Abstract: Journal of Topology and Analysis, Ahead of Print. A subset [math] of a group [math] invariably generates [math] if [math] is generated by [math] for any choice of [math]. A topological group [math] is said to be [math] if it is invariably generated by some subset [math], and [math] if it is topologically invariably generated by some subset [math]. In this paper, we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees. Our main results show that the Lie group [math] and the automorphism group of a regular tree are [math], and that the groups [math] are not [math] for countable fields of infinite transcendence degree over a prime field. Citation: Journal of Topology and Analysis PubDate: 2020-07-21T07:00:00Z DOI: 10.1142/S1793525320500508
Authors:Qayum Khan Pages: 1 - 9 Abstract: Journal of Topology and Analysis, Ahead of Print. Let [math] be a matrix group. Topological [math]-manifolds with Palais-proper action have the [math]-homotopy type of countable [math]-CW complexes (3.2). This generalizes Elfving’s dissertation theorem for locally linear [math]-manifolds (1996). Also, we improve the Bredon–Floyd theorem from compact Lie groups [math] to arbitrary Lie groups [math]. Citation: Journal of Topology and Analysis PubDate: 2020-07-20T07:00:00Z DOI: 10.1142/S179352532050051X
Authors:R. Frigerio Pages: 1 - 17 Abstract: Journal of Topology and Analysis, Ahead of Print. Let [math] be a topological space admitting an amenable cover of multiplicity [math]. We show that, for every [math] and every [math], the image of [math] in the [math]-homology module [math] vanishes. This strengthens previous results by Gromov and Ivanov, who proved, under the same assumptions, that the [math]-seminorm of [math] vanishes. Citation: Journal of Topology and Analysis PubDate: 2020-07-20T07:00:00Z DOI: 10.1142/S1793525320500521
Authors:Asaf Hadari Pages: 1 - 15 Abstract: Journal of Topology and Analysis, Ahead of Print. Let [math] be either the mapping class group of a closed surface of genus [math], or the automorphism group of a free group of rank [math]. Given any homological representation [math] of [math] corresponding to a finite cover, and any term [math] of the Johnson filtration, we show that [math] has finite index in [math], the Torelli subgroup of [math]. Since [math] for [math], this implies for instance that no such representation is faithful. Citation: Journal of Topology and Analysis PubDate: 2020-07-06T07:00:00Z DOI: 10.1142/S1793525320500533
Authors:Julius Lang Pages: 1 - 10 Abstract: Journal of Topology and Analysis, Ahead of Print. We prove that on a surface of negative Euler characteristic, two real-analytic Finsler metrics have the same unparametrized oriented geodesics, if and only if they differ by a scaling constant and addition of a closed 1-form. Citation: Journal of Topology and Analysis PubDate: 2020-06-19T07:00:00Z DOI: 10.1142/S1793525320500491
Authors:Michael Usher Pages: 1 - 56 Abstract: Journal of Topology and Analysis, Ahead of Print. Following proposals of Ostrover and Polterovich, we introduce and study “coarse” and “fine” versions of a symplectic Banach–Mazur distance on certain open subsets of [math] and other open Liouville domains. The coarse version declares two such domains to be close to each other if each domain admits a Liouville embedding into a slight dilate of the other; the fine version, which is similar to the distance on subsets of cotangent bundles of surfaces recently studied by Stojisavljević and Zhang, imposes an additional requirement on the images of these embeddings that is motivated by the definition of the classical Banach–Mazur distance on convex bodies. Our first main result is that the coarse and fine distances are quite different from each other, in that there are sequences that converge coarsely to an ellipsoid but diverge to infinity with respect to the fine distance. Our other main result is that, with respect to the fine distance, the space of star-shaped domains in [math] admits quasi-isometric embeddings of [math] for every finite dimension [math]. Our constructions are obtained from a general method of constructing [math]-dimensional Liouville domains whose boundaries have Reeb dynamics determined by certain autonomous Hamiltonian flows on a given [math]-dimensional Liouville domain. The bounds underlying our main results are proven using filtered equivariant symplectic homology via methods from [J. Gutt and M. Usher, Symplectically knotted codimension-zero embeddings between domains in [math], Duke Math. J. 168 (2019) 2299–2363]. Citation: Journal of Topology and Analysis PubDate: 2020-04-18T07:00:00Z DOI: 10.1142/S179352532050048X
Authors:Aristotelis Panagiotopoulos, Sławomir Solecki Pages: 1 - 27 Abstract: Journal of Topology and Analysis, Ahead of Print. We represent the universal Menger curve as the topological realization [math] of the projective Fraïssé limit [math] of the class of all finite connected graphs. We show that [math] satisfies combinatorial analogues of the Mayer–Oversteegen–Tymchatyn homogeneity theorem and the Anderson–Wilson projective universality theorem. Our arguments involve only [math]-dimensional topology and constructions on finite graphs. Using the topological realization [math], we transfer some of these properties to the Menger curve: we prove the approximate projective homogeneity theorem, recover Anderson’s finite homogeneity theorem, and prove a variant of Anderson–Wilson’s theorem. The finite homogeneity theorem is the first instance of an “injective” homogeneity theorem being proved using the projective Fraïssé method. We indicate how our approach to the Menger curve may extend to higher dimensions. Citation: Journal of Topology and Analysis PubDate: 2020-03-16T07:00:00Z DOI: 10.1142/S1793525320500478
Authors:Koen van den Dungen Pages: 1 - 35 Abstract: Journal of Topology and Analysis, Ahead of Print. We study the Kasparov product on (possibly non-compact and incomplete) Riemannian manifolds. Specifically, we show on a submersion of Riemannian manifolds that the tensor sum of a regular vertically elliptic operator on the total space and an elliptic operator on the base space represents the Kasparov product of the corresponding classes in [math]-theory. This construction works in general for symmetric operators (i.e. without assuming self-adjointness), and extends known results for submersions with compact fibers. The assumption of regularity for the vertically elliptic operator is not always satisfied, but depends on the topology and geometry of the submersion, and we give explicit examples of non-regular operators. We apply our main result to obtain a factorization in unbounded [math]-theory of the fundamental class of a Riemannian submersion, as a Kasparov product of the shriek map of the submersion and the fundamental class of the base manifold. Citation: Journal of Topology and Analysis PubDate: 2020-02-19T08:00:00Z DOI: 10.1142/S1793525320500454
Authors:Qingnan An, George A. Elliott, Zhiqiang Li, Zhichao Liu Pages: 1 - 20 Abstract: Journal of Topology and Analysis, Ahead of Print. In this paper, using ordered total K-theory, we give a K-theoretic classification for the real rank zero inductive limits of direct sums of generalized dimension drop interval algebras. Citation: Journal of Topology and Analysis PubDate: 2020-02-18T08:00:00Z DOI: 10.1142/S1793525320500466
Authors:Adrien Boyer Pages: 1 - 50 Abstract: Journal of Topology and Analysis, Ahead of Print. We investigate properties of some spherical functions defined on hyperbolic groups using boundary representations on the Gromov boundary endowed with the Patterson–Sullivan measure class. We prove sharp decay estimates for spherical functions as well as spectral inequalities associated with boundary representations. This point of view on the boundary allows us to view the so-called property RD (also called Haagerup’s inequality) as a particular case of a more general behavior of spherical functions on hyperbolic groups. We also prove that the family of boundary representations studied in this paper, which can be regarded as a one parameter deformation of the boundary unitary representation, are slow growth representations acting on a Hilbert space admitting a proper 1-cocycle. Citation: Journal of Topology and Analysis PubDate: 2020-01-28T08:00:00Z DOI: 10.1142/S1793525320500429
Authors:J. A. Moya-Pérez, J. J. Nuño-Ballesteros Pages: 1 - 18 Abstract: Journal of Topology and Analysis, Ahead of Print. We show that a 1-parameter family of real analytic map germs [math] with isolated instability is topologically trivial if it is excellent and the family of double point curves [math] in [math] is topologically trivial. In particular, we deduce that [math] is topologically trivial when the Milnor number [math] is constant. Citation: Journal of Topology and Analysis PubDate: 2020-01-28T08:00:00Z DOI: 10.1142/S1793525320500430
Authors:Damian Sawicki, Jianchao Wu Pages: 1 - 25 Abstract: Journal of Topology and Analysis, Ahead of Print. We provide the converses to two results of Roe [Warped cones and property A, Geom. Topol. 9 (2005) 163–178, https://doi.org/10.2140/9t.2005.9.163]: first, the warped cone associated to a free action of an a-T-menable group admits a fibered coarse embedding into a Hilbert space, and second, a free action yielding a warped cone with property A must be amenable. We construct examples showing that in both cases the freeness assumption is necessary. The first equivalence is obtained also for other classes of Banach spaces, in particular for [math]-spaces. Citation: Journal of Topology and Analysis PubDate: 2020-01-15T08:00:00Z DOI: 10.1142/S179352532050034X
Authors:Reid Monroe Harris Pages: 1 - 15 Abstract: Journal of Topology and Analysis, Ahead of Print. We consider the parameter space [math] of smooth plane curves of degree [math]. The universal smooth plane curve of degree [math] is a fiber bundle [math] with fiber diffeomorphic to a surface [math]. This bundle gives rise to a monodromy homomorphism [math], where [math] is the mapping class group of [math]. The main result of this paper is that the kernel of [math] is isomorphic to [math], where [math] is a free group of countably infinite rank. In the process of proving this theorem, we show that the complement [math] of the hyperelliptic locus [math] in Teichmüller space [math] has the homotopy type of an infinite wedge of spheres. As a corollary, we obtain that the moduli space of plane quartic curves is aspherical. The proofs use results from the Weil–Petersson geometry of Teichmüller space together with results from algebraic geometry. Citation: Journal of Topology and Analysis PubDate: 2020-01-15T08:00:00Z DOI: 10.1142/S1793525320500375
Authors:Yongle Jiang Pages: 1 - 35 Abstract: Journal of Topology and Analysis, Ahead of Print. By the work of Brodzki–Niblo–Nowak–Wright and Monod, topological amenability of a continuous group action can be characterized using uniformly finite homology groups or bounded cohomology groups associated to this action. We show that (certain variations of) these groups are invariants for topologically free actions under continuous orbit equivalence. Citation: Journal of Topology and Analysis PubDate: 2020-01-08T08:00:00Z DOI: 10.1142/S1793525320500405
Authors:Valeriia Gladkova, Verna Shum Pages: 1 - 13 Abstract: Journal of Topology and Analysis, Ahead of Print. We continue the exploration of the relationship between conformal dimension and the separation profile by computing the separation of families of spheres in hyperbolic graphs whose boundaries are standard Sierpiński carpets and Menger sponges. In all cases, we show that the separation of these spheres is [math] for some [math] which is strictly smaller than the conformal dimension, in contrast to the case of rank 1 symmetric spaces of dimension [math]. The value of [math] obtained naturally corresponds to a previously known lower bound on the conformal dimension of the associated fractal. Citation: Journal of Topology and Analysis PubDate: 2020-01-07T08:00:00Z DOI: 10.1142/S1793525320500417
Authors:Jinpeng Lu Pages: 1 - 35 Abstract: Journal of Topology and Analysis, Ahead of Print. I prove that the spectrum of the Laplace–Beltrami operator with the Neumann boundary condition on a compact Riemannian manifold with boundary admits a fast approximation by the spectra of suitable graph Laplacians on proximity graphs on the manifold, and similar graph approximation works for metric-measure spaces glued out of compact Riemannian manifolds of the same dimension. Citation: Journal of Topology and Analysis PubDate: 2020-01-07T08:00:00Z DOI: 10.1142/S1793525320500442
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