Journal of Topology and Analysis

For an element in $BS(1,n)=〈t,a\mid ta{t}^{-1}=a^n〉$ written in the normal form $t^{-u}{a}^v{t}^w$ with $u,w\ge 0$ and $v\in \mathbb{Z}$, we exhibit a geodesic word representing the element and give a formula for its word length with respect to the generating set $\{t,a\}$. Using this word length formula, we prove that there are sets of elements of positive density of positive, negative and zero conjugation curvature, as defined by Bar Natan, Duchin and Kropholler.

In this paper, we introduce and study a notion of asymptotic expansion in measure for measurable actions. This generalizes expansion in measure and provides a new perspective on the classical notion of strong ergodicity. Moreover, we obtain structure theorems for asymptotically expanding actions, showing that they admit exhaustions by domains of expansion. As an application, we recover a recent result of Marrakchi, characterizing strong ergodicity in terms of local spectral gaps. We also show that homogeneous strongly ergodic actions are always expanding in measure and establish a connection between asymptotic expansion in measure and asymptotic expanders by means of approximating spaces.

In this paper, we show that the quantum disk, i.e. the quantum space corresponding to the Toeplitz ${\mathrm{\text{C}}}^{*}$-algebra does not admit any compact quantum group structure. We prove that if such a structure existed the resulting compact quantum group would simultaneously be of Kac type and not of Kac type. The main tools used in the solution come from the theory of operators on Hilbert spaces.

Let $M_{\mathrm{\Sigma }}$ be an $n$-dimensional Thom–Mather stratified space of depth $1$. We denote by $\beta M$ the singular locus and by $L$ the associated link. In this paper, we study the problem of when such a space can be endowed with a wedge metric of positive scalar curvature. We relate this problem to recent work on index theory on stratified spaces, giving first an obstruction to the existence of such a metric in terms of a wedge $\alpha$-class ${\alpha }_w(M_{\mathrm{\Sigma }})\in {\mathrm{\text{KO}}}_n$. In order to establish a sufficient condition, we need to assume additional structure: we assume that the link of $M_{\mathrm{\Sigma }}$ is a homogeneous space of positive scalar curvature, $L=G/K$, where the semisimple compact Lie group $G$ acts transitively on $L$ by isometries. Examples of such manifolds include compact semisimple Lie groups and Riemannian symmetric spaces of compact type. Under these assumptions, when $M_{\mathrm{\Sigma }}$ and $\beta M$ are spin, we reinterpret our obstruction in terms of two $\alpha$-classes associated to the resolution of $M_{\mathrm{\Sigma }}$, $M$, and to the singular locus $\beta M$. Finally, when $M_{\mathrm{\Sigma }}$, $\beta M$, $L$ and $G$ are simply connected and $dimM$ is big enough, and when some other conditions on $L$ (satisfied in a large number of cases) hold, we establish the main result of this paper, showing that the vanishing of these two $\alpha$-classes is also sufficient for the existence of a well-adapted wedge metric of positive scalar curvature.

For an orientable surface of finite type equipped with a flat metric with holonomy of finite order $q$, the set of maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when $q\le 2$ is well-studied as such surfaces are (semi-)translation surfaces. Not only is the set always infinite, the core curves form an infinite diameter subset of the curve complex. In this paper we focus on the case $q\ge 3$ and construct examples illustrating a range of behaviors for the embedded cylinder curves. We prove that if $q\ge 3$ and the surface is fully punctured, then the embedded cylinder curves form a finite diameter subset of the curve complex. The same analysis shows that the embedded cylinder curves can only have infinite diameter when the metric has a very specific form. Using this we characterize precisely when the embedded cylinder curves accumulate on a point in the Gromov boundary.

We study the nonlinear embeddability of Banach spaces and the equi-embeddability of the family of Kalton’s interlaced graphs ${({[\mathbb{N}]}^k,d_{𝕂})}_k$ into dual spaces. Notably, we define and study a modification of Kalton’s property ${𝒬}$ that we call property ${{𝒬}}_p$ (with $p\in (1,+\infty ]$). We show that if ${({[\mathbb{N}]}^k,d_{𝕂})}_k$ equi-coarse Lipschitzly embeds into $X^{*}$, then the Szlenk index of $X$ is greater than $\omega$, and that this is optimal, i.e. there exists a separable dual space $Y^{*}$ that contains ${({[\mathbb{N}]}^k,d_{𝕂})}_k$ equi-Lipschitzly and so that $Y$ has Szlenk index ${\omega }^2$. We prove that $c_0$ does not coarse Lipschitzly embed into a separable dual space by a map with distortion strictly smaller than $\frac{3}{2}$. We also show that neither $c_0$ nor $L_1$ coarsely embeds into a separable dual by a weak-to-weak${*}^{}$ sequentially continuous map.

We study the homotopy type of the space of the unitary group ${\mathrm{\text{U}}}_1(C_u^{*}(\left|{\mathbb{Z}}^n\right|))$ of the uniform Roe algebra $C_u^{*}(|{\mathbb{Z}}^n|)$ of ${\mathbb{Z}}^n$. We show that the stabilizing map ${\mathrm{\text{U}}}_1(C_u^{*}(\left|{\mathbb{Z}}^n\right|))\to {\mathrm{\text{U}}}_{\infty }(C_u^{*}(\left|{\mathbb{Z}}^n\right|))$ is a homotopy equivalence. Moreover, when $n=1,2$, we determine the homotopy type of ${\mathrm{\text{U}}}_1(C_u^{*}(\left|{\mathbb{Z}}^n\right|))$, which is the product of the unitary group ${\mathrm{\text{U}}}_1(C^{*}(\left|{\mathbb{Z}}^n\right|))$ (having the homotopy type of ${\mathrm{\text{U}}}_{\infty }(ℂ)$ or $\mathbb{Z}\times B{\mathrm{\text{U}}}_{\infty }(ℂ)$ depending on the parity of $n$) of the Roe algebra $C^{*}(|{\mathbb{Z}}^n|)$ and rational Eilenberg–MacLane spaces.

For any $g\ge 3$, we construct genus-$g$ Lefschetz fibrations over the two-sphere whose slopes are arbitrarily close to $2$. The total spaces of the Lefschetz fibrations can be chosen to be minimal and simply connected. It is also shown that the infimum and the supremum of slopes of all Lefschetz fibrations over the two-sphere are not realized as slopes.

In this paper, we study the moduli space of frozen planet orbits in the Helium atom for an interpolation between instantaneous and mean interactions and show that this moduli space is compact.

We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver [A von Neumann algebra approach to quantum metrics, Mem. Am. Math. Soc. 215(1010) (2012) 1–80]. We show that quantum asymptotic dimension behaves well with respect to metric quotients and direct sums, and is preserved under quantum coarse embeddings. Moreover, we prove that a quantum metric space that equi-coarsely contains a sequence of expanders must have infinite asymptotic dimension. This is done by proving a quantum version of a vertex-isoperimetric inequality for expanders, based upon a previously known edge-isoperimetric one from [K. Temme, M. J. Kastoryano, M. B. Ruskai, M. M. Wolf and F. Verstraete, The ${\chi }^2$-divergence and mixing times of quantum Markov processes, J. Math. Phys. 51(12) (2010) 122201].