Journal of Topology and Analysis

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.

We classify closed, spin${}^{+}$, topological $4$-manifolds with fundamental group $\pi$ of cohomological dimension $\le 3$ (up to $s$-cobordism), after stabilization by connected sum with at most $b_3(\pi )$ copies of $S^2\times S^2$. In general, we must also assume that $\pi$ 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 right-angled Artin groupswhose defining graphs have no $4$-cliques.

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.

For a finitely generated group $G$ acting on a metric space $X$, Roe defined the warped space $X_G$, 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 such actions the group $G$ can be recovered as an appropriately defined automorphism group $Aut(X/X_G)$ when $X$ satisfies a large scale connectedness condition. We show that for a coarsely discontinuous action of a countable group $G$ on a discrete bounded geometry metric space $X$ there is a relation between the maximal Roe algebras of $X$ and $X_G$, namely that there is a ∗-isomorphism $C_{max}^{\ast }(X_G)/{𝒦}\cong (C_{max}^{\ast }(X)/{𝒦})⋊G$, where ${𝒦}$ is the ideal of compact operators. If $X$ has Property A and $G$ is amenable, then $X_G$ has Property A, and thus the maximal Roe algebra and full crossed product can be replaced by the usual Roe algebra and reduced crossed product respectively in the above equation.

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 $\left|{ℒ}\right|$ 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 $\left|{ℒ}\right|$. The obstructions that prevent a simple closed curve in $C$ from being a vanishing cycle are encoded by the adjoint line bundle $K_X\otimes {ℒ}$. In this paper, we consider the linear systems carrying the two simplest types of obstruction. Geometrically, these obstructions manifest on $C$ respectively as an hyperelliptic involution and as a spin structure. In both cases, we determine all the vanishing cycles by investigating the associated monodromy maps, whose target space is the mapping class group $\mathrm{\text{MCG}}(C)$. We show that the image of the monodromy is the subgroup of $\mathrm{\text{MCG}}(C)$ preserving respectively the hyperelliptic involution and the spin structure. The results obtained here support Conjecture $1$ in [R. Crétois and L. Lang, The vanishing cycles of curves in toric surfaces, I, preprint (2017), arXiv:1701.00608] aiming to describe all the vanishing cycles for any pair $(X,{ℒ})$.

Let $\text{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $g\ge 2$, and let $f\in \text{Mod}(S_g)$ be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on $S_g$ that realizes $f$ as an isometry. In other words, this procedure yields an explicit solution to the Nielsen realization problem for cyclic subgroups of $\text{Mod}(S_g)$. Furthermore, we give a purely combinatorial perspective by showing how certain finite order mapping classes can be viewed as fat graph automorphisms. As an application of our realizations, we determine the sizes of maximal reduction systems for certain finite order mapping classes. Moreover, we describe a method to compute the image of finite order mapping classes and the roots of Dehn twists, under the symplectic representation $\mathrm{\Psi }:\text{Mod}(S_g)\to \text{Sp}(2g;\mathbb{Z})$.

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 $A\ll d$ then $D$ can be filled by a homotopy ${\gamma }_t$ with $\left|{\gamma }_t\right|$ bounded by $L+2d+O(\sqrt{A})$.

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