For a nonorientable surface, the twist subgroup is an index 2 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.

We prove that for a free product G with free factor system 𝒢, any automorphism ϕ preserving 𝒢, atoroidal (in a sense relative to 𝒢) and none of whose power send two different conjugates of subgroups in 𝒢 on conjugates of themselves by the same element, gives rise to a semidirect product G⋊ϕℤ that is relatively hyperbolic with respect to suspensions of groups in 𝒢. We recover a theorem of Gautero–Lustig

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 KO∗(ℝ) 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

Let G be a group acting properly and by isometries on a metric space X; it follows that the quotient or orbit space X/G is also a metric space. We study the Vietoris–Rips and Čech complexes of X/G. 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

Let G be a semisimple Lie group with finite center, K⊂G a maximal compact subgroup, and P⊂G a parabolic subgroup. Following ideas of P. Y. Gaillard, one may use G-invariant differential forms on G/K×G/P to construct G-equivariant Poisson transforms mapping differential forms on G/P to differential forms on G/K. Such invariant forms can be constructed using finite-dimensional representation theory.

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

Let Γ be either the mapping class group of a closed surface of genus ≥2, or the automorphism group of a free group of rank ≥3. Given any homological representation ρ of Γ corresponding to a finite cover, and any term ℐk of the Johnson filtration, we show that ρ(ℐk) has finite index in ρ(ℐ), the Torelli subgroup of Γ. Since [ℐ:ℐk]=∞ for k>1, this implies for instance that no such representation is faithful

A subset S of a group G invariably generates G if G is generated by {sg(s)|s∈S} for any choice of g(s)∈G,s∈S. A topological group G is said to be ℐ𝒢 if it is invariably generated by some subset S⊆G, and 𝒯ℐ𝒢 if it is topologically invariably generated by some subset S⊆G. In this paper, we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees

Let G be a matrix group. Topological G-manifolds with Palais-proper action have the G-homotopy type of countable G-CW complexes (3.2). This generalizes Elfving’s dissertation theorem for locally linear G-manifolds (1996). Also, we improve the Bredon–Floyd theorem from compact Lie groups G to arbitrary Lie groups G.

Let X be a topological space admitting an amenable cover of multiplicity k∈ℕ. We show that, for every n≥k and every α∈Hn(X;ℝ), the image of α in the ℓ1-homology module Hnℓ1(X;ℝ) vanishes. This strengthens previous results by Gromov and Ivanov, who proved, under the same assumptions, that the ℓ1-seminorm of α vanishes.

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.

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 ℂn 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

We represent the universal Menger curve as the topological realization |𝕄| of the projective Fraïssé limit 𝕄 of the class of all finite connected graphs. We show that 𝕄 satisfies combinatorial analogues of the Mayer–Oversteegen–Tymchatyn homogeneity theorem and the Anderson–Wilson projective universality theorem. Our arguments involve only 0-dimensional topology and constructions on finite graphs

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 KK-theory. This construction works in general for symmetric

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.

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

We show that a 1-parameter family of real analytic map germs ft:(ℝ2,0)→(ℝ3,0) with isolated instability is topologically trivial if it is excellent and the family of double point curves D(ft) in (ℝ2,0) is topologically trivial. In particular, we deduce that ft is topologically trivial when the Milnor number μ(D(ft)) is constant.

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

We consider the parameter space 𝒰d of smooth plane curves of degree d. The universal smooth plane curve of degree d is a fiber bundle ℰd→𝒰d with fiber diffeomorphic to a surface Σg. This bundle gives rise to a monodromy homomorphism ρd:π1(𝒰d)→Mod(Σg), where Mod(Σg):=π0(Diff+(Σg)) is the mapping class group of Σg. The main result of this paper is that the kernel of ρ4:π1(𝒰4)→Mod(Σ3) is isomorphic

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.

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 nd−1d for some d which is strictly smaller than the conformal dimension, in contrast to the case

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.

We constructed infinitely many involutions on M×M with fixed point set homeomorphic to M. All these involutions are ℤ/2-isovariantly homotopy equivalent to interchange and are inequivalent to one another in the TOP category.

For a finitely generated group G acting on a metric space X, Roe defined the warped space XG, which one can view as a kind of large scale quotient of X by the action of G. In this paper, we generalize this notion to the setting of actions of arbitrary groups on large scale spaces. We then restrict our attention to what we call coarsely discontinuous actions by coarse equivalences and show that for

Several formulas for computing coarse indices of twisted Dirac type operators are introduced. One type of such formulas is by composition product in E-theory. The other type is by module multiplications in K-theory, which also yields an index theoretic interpretation of the duality between Roe algebra and stable Higson corona.

We resume the study initiated in [R. Crétois and L. Lang, The vanishing cycles of curves in toric surfaces, I, preprint (2017), arXiv:1701.00608]. For a generic curve C in an ample linear system |ℒ| on a toric surface X, a vanishing cycle of C is an isotopy class of simple closed curve that can be contracted to a point along a degeneration of C to a nodal curve in |ℒ|. The obstructions that prevent

We classify closed, spin+, topological 4-manifolds with fundamental group π of cohomological dimension ≤3 (up to s-cobordism), after stabilization by connected sum with at most b3(π) copies of S2×S2. In general, we must also assume that π satisfies certain K-theory and assembly map conditions. Examples for which these conditions hold include the torsion-free fundamental groups of 3-manifolds and all

We find all global quasiconformal maps (with respect to the Carnot metric) on a particular 2-step Carnot group. In particular, all the global quasiconformal maps of this Carnot group permute the left cosets of the center, verifying a conjecture by Xie for this particular case.

Let Mod(Sg) denote the mapping class group of the closed orientable surface Sg of genus g≥2, and let f∈Mod(Sg) be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on Sg that realizes f as an isometry. In other words, this procedure yields an explicit solution to the Nielsen realization problem for cyclic subgroups of Mod(Sg). Furthermore, we give a purely

We answer a question of Liokumovich–Nabutovsky–Rotman showing that if D is a Riemannian 2-disk with boundary length L, diameter d and area A≪d then D can be filled by a homotopy γt with |γt| bounded by L+2d+O(A).

We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2,ℝ) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain

In this paper, we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behavior of the Betti numbers in the lower model is characterized by the notion of critical dimension, which was introduced by Costa and Farber in [Large random simplicial complexes

We investigate projective properties of Lorentzian surfaces. In particular, we prove that if T is a non-flat torus, then the index of its isometry group in its projective group is at most two. We also prove that any topologically finite non-compact surface can be endowed with a metric having a non-isometric projective transformation of infinite order.

An area-preserving diffeomorphism of an annulus has an “action function” which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit. If an area-preserving annulus diffeomorphism

We prove that every proper n-dimensional length metric space admits an “approximate isometric embedding” into Lorentzian space ℝ3n+6,1. By an “approximate isometric embedding” we mean an embedding which preserves the energy functional on a prescribed set of geodesics connecting a dense set of points.

We construct a family of trivial 2-knots ki in ℝ4 such that the maximal complexity of 2-knots in any isotopy connecting ki with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of ki. Here, we can either construct ki as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or construct PL-knots ki, consider isotopies

For a given quasi-Fuchsian representation ρ:π1(S)→PSL2ℂ of the fundamental group of a closed surface S of genus g≥2, we prove that a generic branched complex projective structure on S with holonomy ρ and two branch points can be obtained from some unbranched structure on S with the same holonomy by bubbling, i.e. a suitable connected sum with a copy of ℂℙ1.

We show that the diameter of the skinning map of an acylindrical hyperbolic 3-manifold M is bounded on 𝜀-thick Teichmüller geodesics by a constant depending only on 𝜀 and the topological type of ∂M.

We apply quantitative (or controlled) K-theory to prove that a certain Lp assembly map is an isomorphism for p∈[1,∞) when an action of a countable discrete group Γ on a compact Hausdorff space X has finite dynamical complexity. When p=2, this is a model for the Baum–Connes assembly map for Γ with coefficients in C(X), and was shown to be an isomorphism by Guentner et al.

Different diffeomorphisms can give the same C∗ crossed product algebra. Our main purpose is to show that we can still classify dynamical systems with some appropriate smooth crossed product algebras when their corresponding C∗ crossed product algebras are isomorphic. For this purpose, we construct two minimal unique ergodic diffeomorphisms α and β of S3×S6×S8. The C∗ algebras classification theory

For X a metric space and r>0 a scale parameter, the Vietoris–Rips simplicial complex VR<(X;r) (resp. VR≤(X;r)) has X as its vertex set, and a finite subset σ⊆X as a simplex whenever the diameter of σ is less than r (resp. at most r). Though Vietoris–Rips complexes have been studied at small choices of scale by Hausmann and Latschev [13,16], they are not well-understood at larger scale parameters. In

A rational projective plane (ℚℙ2) is a simply connected, smooth, closed manifold M such that H∗(M;ℚ)≅ℚ[α]/〈α3〉. An open problem is to classify the dimensions at which such a manifold exists. The Barge–Sullivan rational surgery realization theorem provides necessary and sufficient conditions that include the Hattori–Stong integrality conditions on the Pontryagin numbers. In this paper, we simplify these

We show that a uniform probability measure supported on a specific set of piecewise linear loops in a nontrivial free homotopy class in a multi-punctured plane is overwhelmingly concentrated around loops of minimal lengths. Our approach is based on extending Mogulskii’s theorem to closed paths, which is a useful result of independent interest. In addition, we show that the above measure can be sampled

We introduce the notion of regular finite decomposition complexity of a metric family. This generalizes Gromov’s finite asymptotic dimension and is motivated by the concept of finite decomposition complexity (FDC) due to Guentner, Tessera and Yu. Regular finite decomposition complexity implies FDC and has all the permanence properties that are known for FDC, as well as a new one called Finite Quotient

We consider two disjoint and homotopic non-contractible embedded loops on a Riemann surface and prove the existence of a non-contractible orbit for a Hamiltonian function on the surface whenever it is sufficiently large on one of the loops and sufficiently small on the other. This gives the first example of an estimate from above for a generalized form of the Biran–Polterovich–Salamon capacity for

In a compact geodesic metric space of topological dimension one, the minimal length of a loop in a free homotopy class is well-defined, and provides a function l:π1(X)→ℝ+∪{∞} (the value ∞ being assigned to loops which are not freely homotopic to any rectifiable loops). This function is the marked length spectrum. We introduce a subset Conv(X), which is the union of all non-constant minimal loops of

In this paper we provide a framework for the study of isoperimetric problems in finitely generated groups, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions, one can restrict to simplicial spheres of particular shapes, called “round” and “unfolded”, provided that a bounded quasi-geodesic combing exists. We prove that the

It is known that the genus two surface admits a piecewise flat metric with conical singularities which is extremal for the systolic inequality among all nonpositively curved metrics. We prove that this piecewise flat metric is also critical for slow metric variations, without curvature restrictions, for another type of systolic inequality involving the lengths of the shortest noncontractible loops

We investigate to what extent a nilpotent Lie group is determined by its C∗-algebra. We prove that, within the class of exponential Lie groups, direct products of Heisenberg groups with abelian Lie groups are uniquely determined even by their unitary dual, while nilpotent Lie groups of dimension ≤5 are uniquely determined by the Morita equivalence class of their C∗-algebras. We also find that this

Given a discrete group G, for any integer r≥0 we consider the family of all virtually abelian subgroups of G of rank at most r. We give an upper bound for the Bredon cohomological dimension of G for this family for a certain class of groups acting on CAT(0) spaces. This covers the case of Coxeter groups, Right-angled Artin groups, fundamental groups of special cube complexes and graph products of finite

We study basic Dolbeault cohomology and find new Weitzenböck formulas on a transversely Kähler foliation. We investigate conditions on mean curvature and Ricci curvature that impose restrictions on basic Dolbeault cohomology. For example, we prove that on a transversely Kähler foliation with positive transversal Ricci curvature, there are no nonzero basic-harmonic forms of type (r,0), among other results

In this note, we define a distance between two pointed locally integral current spaces. We prove that a sequence of pointed locally integral current spaces converges with respect to this distance if and only if it converges in the sense of Lang–Wenger. This enables us to state the compactness theorem by Lang–Wenger for pointed locally integral current spaces in terms of a distance function.

In this paper, we use the G-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe the spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtle sign terms in the G-spin theorem for an isometry, with isolated fixed points,

Let Γ be the fundamental group of a surface of finite type and Comm(Γ) be its abstract commensurator. Then Comm(Γ) contains the solvable Baumslag–Solitar groups 〈a,b:aba−1=bn〉 for any n>1. Moreover, the Baumslag–Solitar group 〈a,b:ab2a−1=b3〉 has an image in Comm(Γ) that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of Comm(Γ)

An ATAI (or ATAF, respectively) algebra, introduced in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal.3 (2011) 385–404] (or in [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal.256 (2009) 3861–3891], respectively) is an inductive limit limn→∞(An=⊕i=1Ani

Let BPUn be the classifying space of PUn, the projective unitary group of order n, for n>1. We use a Serre spectral sequence to determine the ring structure of H∗(BPUn;ℤ) up to degree 10, as well as a family of distinguished elements of H2p+2(BPUn;ℤ), for each prime divisor p of n. We also study the primitive elements of H∗(BUn;ℤ) as a comodule over H∗(K(ℤ,2);ℤ), where the comodule structure is given

We show that any stable exponentially harmonic map from a compact Riemannian manifold into a compact simply-connected δ-pinched Riemannian manifold under certain circumstance is constant in two different versions. We also prove that a non-constant exponentially harmonic map from a compact hypersurface into a compact Riemannian manifold satisfying certain condition is unstable.

We study isometric actions of Steinberg groups on Hadamard manifolds. We prove some rigidity properties related to these actions. In particular, we show that every isometric action of Stn(𝔽p〈t1,…,tk〉) on Hadamard manifold when n≥3 factors through a finite quotient. We further study actions on infinite-dimensional manifolds and prove a fixed-point theorem related to such actions.

In this paper, we give a close-to-sharp answer to the basic questions: When is there a continuous way to add a point to a configuration of n ordered points on a surface S of finite type so that all the points are still distinct? When this is possible, what are all the ways to do it? More precisely, let PConfn(S) be the space of ordered n-tuple of distinct points in S. Let fn(S):PConfn+1(S)→PConfn(S)

We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion-free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other