Projection complexes and quasimedian maps

Acylindrical actions on projection complexes

We simplify the construction of projection complexes due to Bestvina-Bromberg-Fujiwara. To do so, we introduce a sharper version of the Behrstock inequality, and show that it can always be enforced. Furthermore, we use the new setup to prove acylindricity results for the action on the projection complexes. We also treat quasi-trees of metric spaces associated to projection complexes, and prove an acylindricity criterion in that context as well.

投影复形理论中的两个新构造

Two new constructions in the theory of projection complexes

本文提出两个新构造，简化投影复形与度量空间拟树的度量结构，并据此给出Hume定理（映射类群拟等距嵌入有限树乘积）的更短证明。

Hyperbolic quotients of projection complexes

This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes. In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex stabilizers is a free product of certain conjugates of these subgroups. In this paper, we study both the quotient of the projection complex by this normal subgroup and the action of the quotient group on the quotient of the projection complex. We show that under certain conditions that the quotient complex is $δ$-hyperbolic. Additionally, under certain circumstan…

Research announcement: A combination theorem for acylindrical complexes of hyperbolic groups and Cannon-Thurston maps

This is an announcement of some of the results obtained as a part of the second author's Ph.D. thesis. In the first part, we prove that the fundamental group of an acylindrical complex of hyperbolic groups with finite edge groups is hyperbolic in which the vertex groups are quasiconvex. In the second part of the article, we prove the existence of Cannon-Thurston maps for certain subcomplexes of groups in acylindrical complexes of hyperbolic groups (see Theorem 0.4).

Injective Simplicial Maps of the Complexes of Curves of Nonorientable Surfaces

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components, and $\mathcal{C}(N)$ be the complex of curves of $N$. Suppose that $g + n \leq 3$ or $g + n \geq 5$. If $λ: \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ is an injective simplicial map, then $λ$ is induced by a homeomorphism of $N$.

Superinjective Simplicial Maps of the Two-sided Curve Complexes on Nonorientable Surfaces

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components with $g \geq 5$, $n \geq 0$. Let $\mathcal{T}(N)$ be the two-sided curve complex of $N$. If $λ:\mathcal{T}(N) \rightarrow \mathcal{T}(N)$ is a superinjective simplicial map, then there exists a homeomorphism $h : N \rightarrow N$ unique up to isotopy such that $H(α) = λ(α)$ for every vertex $α$ in $\mathcal{T}(N)$ where $H=[h]$.