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Yulan Qing

Publications and source records attributed to Yulan Qing.

At least 19 recordsLinked to original sources
arXiv2026

First passage percolation preserves sublinearly Morse boundaries

Sublinearly Morse directions in proper geodesic spaces are defined by sublinearly Morse stability. In this paper we offer an alternative characterization for sublinearly Morse geodesic lines via middle recurrence. We then study first passage percolation (FPP) on proper geodesic graphs of bounded degree. We associate an i.i.d. collection of random passage times to each edge. Under suitable conditions on the passage time distribution, we prove that sublinearly Morse boundaries are invariant under first passage percolation.

math.GT↗
arXiv2026

The non-peripheral curve graph and divergence in big mapping class groups

We introduce a numerical invariant $\zeta(\Sigma)$ measuring the end-complexity of $\Sigma$ and use it to organize coarse-geometric features of Map($\Sigma$). Our main tool is the \emph{non-peripheral curve graph} $C_{\rm np}(\Sigma)$, whose vertices are those essential simple closed curves that cannot be pushed out of every compact subsurface, with edges given by disjointness. Assuming Map($\Sigma$) is CB-generated and $\zeta(\Sigma)\ge 5$, we prove that $C_{\rm np}(\Sigma)$ is connected, has infinite diameter, is Gromov hyperbolic, and that the Map($\Sigma$)-action has unbounded orbits. As applications, we show that if $\zeta(\Sigma)\ge 4$ then Map($\Sigma$) has infinite coarse rank, and if $\zeta(\Sigma)\ge 5$ then Map($\Sigma$) has at most quadratic divergence, hence is one-ended.

math.GT↗
arXiv2025

The number of ends of big mapping class groups

We analyze the number of ends of the mapping class group of a stable avenue surface. We prove that the mapping class group is one-ended whenever the stable avenue surface has at least one end of discrete type. Our method is to show that the associated translatable curve graph, which is quasi-isometric to the mapping class group, is one-ended.

math.GT↗
arXiv2025

First-passage percolation, non-positive curvature, and radial maps

Given an infinite connected graph $G$, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges of the graph, a process called first-passage percolation. Assume that the graph is infinite and of bounded degree. Assume the edge length distribution, $\nu$, has a finite expectation and is supported on $[0, \infty)$. We prove in this paper that non-positive curvature almost surely is not preserved by the associated percolation. In particular, Gromov hyperbolicity and coarse CAT(0) property of graphs are almost surely not preserved. We also show that if a graph contains a Morse geodesic ray, then the resulting image of the ray under first-passage percolation is no longer Morse. Lastly, we show that first-passage percolation almost surely is a radial map on $G$.

math.PR↗
arXiv2025

Sublinear Morse Geodesics and First Passage Percolation

Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges of the graph. Assume that the graph is infinite and of bounded degree. Assume also strict positivity and finite expectation of the edge length distribution and existence of a sublinearly Morse bi-infinite geodesic line, we prove that almost surely there exists a bi-infinite geodesic line. This generalizes a previous result of \cite{BT17} regarding Morse geodesics.

math.GT↗
arXiv2025

Quasi-redirecting boundaries of non-positively curved groups

The quasi-redirecting (QR) boundary is a close generalization of the Gromov boundary to all finitely generated groups. In this paper, we establish that the QR boundary exists as a topological space for several well-studied classes of groups. These include fundamental groups of irreducible non-geometric 3-manifolds, groups that are hyperbolic relative to subgroups with well-defined QR boundaries, right-angled Artin groups whose defining graphs are trees, and right-angled Coxeter groups whose defining flag complexes are planar. This result significantly broadens the known existence of QR boundaries. Additionally, we give a complete characterization of the QR boundaries of Croke-Kleiner admissible groups that act geometrically on CAT(0) spaces. We show that these boundaries are non-Hausdorff and can be understood as one-point compactifications of the Morse-like directions. Finally, we prove that if G is hyperbolic relative to subgroups with well-defined QR boundaries, then the QR boundary of G maps surjectively onto its Bowditch boundary.

math.GR↗
arXiv2024

Topological and Dynamic Properties of the Sublinearly Morse Boundary and the Quasi-Redirecting Boundary

Sublinearly Morse boundaries of proper geodesic spaces are introduced by Qing, Rafi and Tiozzo. Expanding on this work, Qing and Rafi recently developed the quasi-redirecting boundary, denoted $\partial G$, to include all directions of metric spaces at infinity. Both boundaries are topological spaces that consist of equivalence classes of quasi-geodesic rays and are quasi-isometrically invariant. In this paper, we study these boundaries when the space is equipped with a geometric group action. In particular, we show that $G$ acts minimally on $\partial_κG$ and that contracting elements of G induces a weak north-south dynamic on $\partial_κG$. We also prove, when $\partial G$ exists and $|\partial_κG|\geq3$, $G$ acts minimally on $\partial G$ and $\partial G$ is a second countable topological space. The last section concerns the restriction to proper CAT(0) spaces and finite dimensional \CAT cube complexes. We show that when $G$ acts geometrically on a finite dimensional CAT(0) cube complex (whose QR boundary is assumed to exist), then a nontrivial QR boundary implies the existence of a Morse element in $G$. Lastly, we show that if $X$ is a proper cocompact CAT(0) space, then $\partial G$ is a visibility space.

math.MG↗
arXiv2024

The quasi-redirecting Boundary

We generalize the notion of Gromov boundary to a larger class of metric spaces beyond Gromov hyperbolic spaces. Points in this boundary are classes of quasi-geodesic rays and the space is equipped with a topology that is naturally invariant under quasi-isometries. It turns out that this boundary is compatible with other notions of boundary in many ways; it contains the sublinearly Morse boundary as a topological subspace and it matches the Bowditch boundary of relative hyperbolic spaces when the peripheral subgroups have no intrinsic hyperbolicity. We also give a complete description of the boundary of the Croke-Kleiner group where the quasi-redirecting boundary reveals a new class of QI-invariant, Morse-like quasi-geodesics.

math.GR↗
arXiv2024

An algorithm to decide if an outer automorphism is geometric

An outer automorphism of a free group is geometric if it can be represented by a homeomorphism of a compact surface. Bestvina and Handel gave an algorithmic characterization of geometric irreducible outer automorphisms using relative train tracks in 1995. The general case of detecting geometric outer automorphisms remained open, with a few partial results appearing subsequently. In this paper we give a complete resolution to the problem: an algorithm that can decide if a general outer automorphism is geometric. The algorithm is constructive and produces a realizing surface homeomorphism if one exists. We make use of advances in train-track theory, in conjunction with the Guirardel core of tree actions and Nielsen-Thurston theory for surfaces.

math.GR↗
arXiv2023

Sublinear biLipschitz equivalence and sublinearly Morse boundaries

A sublinear biLipschitz equivalence (SBE) between metric spaces is a map from one space to another that distorts distances with bounded multiplicative constants and sublinear additive error. Given any sublinear function $κ$, $κ$-Morse boundaries are defined for all geodesic proper metric spaces as a quasi-isometrically invariant and metrizable topological space of quasi-geodesic rays. In this paper, we prove that $κ$-Morse boundaries of proper geodesic metric spaces are invariant under suitable SBEs. A tool in the proof is the use of sublinear rays, that is, sublinear bilispchitz embeddings of the half line, generalizing quasi-geodesic rays. As an application we distinguish a pair of right-angled Coxeter groups brought up by Behrstock up to sublinear biLipschitz equivalence. We also show that under mild assumptions, generic random walks on countable groups are sublinear rays.

math.GR↗
arXiv2023

Geometry and dynamics on sublinearly Morse boundaries of CAT(0) groups

Given a sublinear function $κ$, $κ$-Morse boundaries $\pka X$ of proper \CAT spaces are introduced by Qing, Rafi and Tiozzo. It is a topological space that consists of a large set of quasi-geodesic rays and it is quasi-isometrically invariant and metrizable. In this paper, we study the sublinearly Morse boundaries with the assumption that there is a proper cocompact action of a group $G$ on the \CAT space in question. We show that $G$ acts minimally on $\pka G$ and that contracting elements of $G$ induces a weak north-south dynamic on $\pka G$. Furthermore, we show that a homeomorphism $f \from \pka G \to \pka G'$ comes from a quasi-isometry if and only if $f$ is successively quasi-m{ö}bius and stable. Lastly, we characterize exactly when the sublinearly Morse boundary of a \CAT space is compact.

math.GR↗
arXiv2022

Genericity of sublinearly Morse directions in CAT(0) spaces and the Teichmüller space

We show that the sublinearly Morse directions in the visual boundary of a rank-1 CAT(0) space with a geometric group action are generic in several commonly studied senses of the word, namely with respect to Patterson-Sullivan measures and stationary measures for random walks. We deduce that the sublinearly Morse boundary is a model of the Poisson boundary for finitely supported random walks on groups acting geometrically on rank-1 CAT (0) spaces. We prove an analogous result for mapping class group actions on Teichmüller space. Our main technical tool is a criterion, valid in any unique geodesic metric space, that says that any geodesic ray with sufficiently many (in a statistical sense) strongly contracting segments is sublinearly contracting.

math.GR↗
arXiv2022

Sublinearly Morse Boundary I: CAT(0) Spaces

To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed that the visual boundary of non-positively curved (CAT(0)) groups is not well-defined, since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. For any sublinear function $κ$, we consider a subset of the visual boundary called the $κ$-Morse boundary and show that it is QI-invariant and metrizable. This is to say, the $κ$-Morse boundary of a CAT(0) group is well-defined. In the case of Right-angled Artin groups, it is shown in the Appendix that the Poisson boundary of random walks is naturally identified with the $\sqrt{t \log t}$--boundary.

math.GT↗
arXiv2022

Sublinearly Morse Boundary of CAT(0) admissible groups

We show that if G is an admissible group acting geometrically on a CAT(0) space X, then G is a hierarchically hyperbolic space and with mild assumptions the sublinearly-Morse boundary of the group is a topological model for associated Poisson boundaries .

math.GR↗
arXiv2020

Quasi-geodesics in Out(F_n) and their shadows in sub-factors

We study the behaviour of quasi-geodesics in Out(F_n). Given an element f in Out(F_n) there are several natural paths connecting the origin to f in Out(F_n); for example, paths associated to sequences of Stallings folds and paths induced by the shadow of greedy folding paths in Outer Space. We show that none of these paths is, in general, a quasi-geodesic in Out(F_n). In fact, in contrast with the mapping class group setting, we construct examples where any quasi-geodesic in Out(F_n) connecting f to the origin will have to back-track in some free factor of F_n.

math.GR↗
arXiv2020

Sublinearly Morse Boundary II: Proper geodesic spaces

We build an analogue of the Gromov boundary for any proper geodesic metric space, hence for any finitely generated group. More precisely, for any proper geodesic metric space $X$ and any sublinear function $κ$, we construct a boundary for $X$, denoted $\mathcal{\partial}_κ X$, that is quasi-isometrically invariant and metrizable. As an application, we show that when $G$ is the mapping class group of a finite type surface, or a relatively hyperbolic group, then with minimal assumptions the Poisson boundary of $G$ can be realized on the $κ$-Morse boundary of $G$ equipped the word metric associated to any finite generating set.

math.GT↗