arXiv subjects

Yulan Qing

Publications and source records attributed to Yulan Qing.

25 records · Page 2Linked to original sources

arXiv2020

Big mapping class groups with hyperbolic actions: classification and applications

We address the question of determining which mapping class groups of infinite-type surfaces admit nonelementary continuous actions on hyperbolic spaces. More precisely, let $Σ$ be a connected, orientable surface of infinite type with tame endspace whose mapping class group is generated by a coarsely bounded subset. We prove that $\mathrm{Map}(Σ)$ admits a continuous nonelementary action on a hyperbolic space if and only if $Σ$ contains a finite-type subsurface which intersects all its homeomorphic translates. When $Σ$ contains such a nondisplaceable subsurface $K$ of finite type, the hyperbolic space we build is constructed from the curve graphs of $K$ and its homeomorphic translates via a construction of Bestvina, Bromberg and Fujiwara. Our construction has several applications: first, the second bounded cohomology of $\mathrm{Map}(Σ)$ contains an embedded $\ell^1$; second, using work of Dahmani, Guirardel and Osin, we deduce that $\mathrm{Map}(Σ)$ contains nontrivial normal free subgroups (while it does not if $Σ$ has no nondisplaceable subsurface of finite type), has uncountably many quotients and is SQ-universal.

math.GT↗

arXiv2019

Geometry of right-angled Coxeter groups on the Croke-Kleiner spaces

In this paper we study the right-angled Coxeter groups that acts geometrically on the Salvetti complex of a certain right-angled Artin group, which we refer to as Croke-Kleiner spaces. We prove that any right-angled Coxeter group that acts geometrically on the Croke-Kleiner spaces acts with $π/2$ angles between reflecting axes, while the quasi-isometric right-angled Artin group can act with angles that are any real number in the range $(0, π/2]$. The contrast between the two examples shows that in this case a right-angled Coxeter group is geometrically more "rigid" than its quasi isometric counterpart.

math.GR↗

arXiv2018

Dilatation of outer automorphisms of Right-angled Artin Groups

We study the dilatation of outer automorphisms of right-angled Artin groups. Given a right-angled Artin group defined on a simplicial graph: $A(Γ) = \langle V | E \rangle$ and an automorphism $ϕ\in Out(A(Γ))$ there is a natural measure of how fast the length of a word of $A(Γ)$ grows after $n$ iterations of $ϕ$ as a function of $n$, which we call the dilatation of $w$ under $ϕ$. We define the dilatation of $ϕ$ as the supremum over dilatations of all $w \in A(Γ)$. Assuming that $ϕ$ is a pure and square map, we show that if the dilatation of $ϕ$ is positive, then either there exists a free abelian special subgroup on which that dilatation is realized; or there exists a strata of either free or free abelian groups on which the dilatation is realized.

math.GR↗

arXiv2018

Excursions of generic geodesics in right-angled Artin groups and graph products

Motivated by the notion of cusp excursion in geometrically finite hyperbolic manifolds, we define a notion of excursion in any subgroup of a given group, and study its asymptotic distribution for right-angled Artin groups and graph products. In particular, for any irreducible right-angled Artin group we show that with respect to the counting measure, the maximal excursion of a generic geodesic in any flat tends to $\log n$, where $n$ is the length of the geodesic. In this regard, irreducible RAAGs behave like a free product of groups. In fact, we show that the asymptotic distribution of excursions detects the growth rate of the RAAG and whether it is reducible.

math.GT↗

arXiv2017

Convexity of balls in the outer space

In this paper we study the convexity properties of geodesics and balls in Outer space equipped with the Lipschitz metric. We introduce a class of geodesics called balanced folding paths and show that, for every loop $α$, the length of $α$ along a balanced folding path is not larger than the maximum of its lengths at the end points. This implies that out-going balls are weakly convex. We then show that these results are sharp by providing several counterexamples.

math.GT↗

arXiv2017

Uniform fellow traveling between surgery paths in the sphere graph

We show that the Hausdorff distance between any forward and any backward surgery paths in the sphere graph is at most 2. From this it follows that the Hausdorff distance between any two surgery paths with the same initial sphere system and same target sphere system is at most 4. Our proof relies on understanding how surgeries affect the Guirardel core associated to sphere systems. We show that applying a surgery is equivalent to performing a Rips move on the Guirardel core.

math.GR↗

arXiv2014

Actions of Right-Angled Coxeter Groups on the Croke Kleiner Spaces

It is an open question whether right-angled Coxeter groups have unique group-equivariant visual boundaries. Croke and Kleiner present a right-angled Artin group with more than one visual boundary. In this paper we present a right-angled Coxeter group with non-unique equivariant visual boundary. The main theorem is that if right-angled Coxeter groups act geometrically on a Croke-Kleiner spaces, then the local angles in those spaces all have to be right angles. We present a specific right-angled Coxeter group with non-unique equivariant visual boundary. However, we conjecture that the right an- gled Coxeter groups that can act geometrically on a given CAT(0) space are far from unique.

math.GR↗