Journal of Topology and Analysis

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 $\partial M$.

Let $\mathrm{\Gamma }$ be the fundamental group of a surface of finite type and $\mathrm{\text{Comm}}(\mathrm{\Gamma })$ be its abstract commensurator. Then $\mathrm{\text{Comm}}(\mathrm{\Gamma })$ contains the solvable Baumslag–Solitar groups $〈a,b:ab{a}^{-1}=b^n〉$ for any $n>1$. Moreover, the Baumslag–Solitar group $〈a,b:a{b}^2{a}^{-1}=b^3〉$ has an image in $\mathrm{\text{Comm}}(\mathrm{\Gamma })$ that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of $\mathrm{\text{Comm}}(\mathrm{\Gamma })$ are concrete objects amenable to computational methods. For example, we give a proof that $〈a,b:a{b}^2{a}^{-1}=b^3〉$ is not residually finite without the use of normal forms of HNN extensions.

We construct a family of trivial $2$-knots $k^i$ in ${ℝ}^4$ such that the maximal complexity of $2$-knots in any isotopy connecting $k^i$ with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of $k^i$.

Here, we can either construct $k^i$ as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or construct PL-knots $k^i$, consider isotopies through PL knots, and measure the complexity of a PL-knot as the minimal number of flat $2$-simplices in its triangulation.

These results contrast with the situation of classical knots in ${ℝ}^3$, where every unknot can be untied through knots of complexity that is only polynomially higher than the complexity of the initial knot.

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.

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 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, of a closed spin hyperbolic 2- or 4-manifold.

Given a discrete group $G$, for any integer $r\ge 0$ 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 $\mathrm{\text{CAT}}(0)$ spaces. This covers the case of Coxeter groups, Right-angled Artin groups, fundamental groups of special cube complexes and graph products of finite groups. Our construction partially answers a question of Lafont.

We investigate to what extent a nilpotent Lie group is determined by its $C^{\ast }$-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 $\le 5$ are uniquely determined by the Morita equivalence class of their $C^{\ast }$-algebras. We also find that this last property is shared by the filiform Lie groups and by the $6$-dimensional free two-step nilpotent Lie group.

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 ${lim}_{n\to \infty }(A_n={\oplus }_{i=1}{A}_n^i,{\varphi }_{nm})$, where each $A_n^i$ is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. 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], the second author classified all ATAI algebras by an invariant consisting orderd total $K$-theory and tracial state spaces of cut down algebras under an extra restriction that all element in $K_1(A)$ are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic $K_1$-group as an addition to the invariant used 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]. The theorem is proved by reducing the class to the classification theorem of ${𝒜}{\mathscr{H}}{𝒟}$ algebras with ideal property which is done in [G. Gong, C. Jiang and L. Li, A classification of inductive limit C*-algebras with ideal property, preprint (2016), arXiv:1607.07681]. Our theorem generalizes the main theorem of [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], [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] (see Corollary 4.3).

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

For a given quasi-Fuchsian representation $\rho :{\pi }_1(S)\to {\mathrm{\text{PSL}}}_2ℂ$ of the fundamental group of a closed surface $S$ of genus $g\ge 2$, we prove that a generic branched complex projective structure on $S$ with holonomy $\rho$ 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 $ℂ{\mathbb{P}}^1$.