scholarly journals Rigidity of the strongly separating curve graph

2016 ◽  
Vol 65 (4) ◽  
pp. 813-832
Author(s):  
Brian Bowditch
Keyword(s):  
Curve Graph

EXHAUSTION OF THE CURVE GRAPH VIA RIGID EXPANSIONS

2018 ◽  
Vol 61 (1) ◽  
pp. 195-230 ◽  
Author(s):  
JESÚS HERNÁNDEZ HERNÁNDEZ
Keyword(s):  
Topological Type ◽  
Orientable Surface ◽  
Finite Topological Type ◽  
Curve Graph ◽  
Finite Set ◽  
Rigid Set

AbstractFor an orientable surfaceSof finite topological type with genusg≥ 3, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph$\mathcal{C}$(S). The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set in Aramayona and Leininger,J. Topology Anal.5(2) (2013), 183–203 and Aramayona and Leininger,Pac. J. Math.282(2) (2016), 257–283, and in fact a consequence of our proof is that Aramayona and Leininger's set also exhausts the curve graph via rigid expansions.

scholarly journals Small intersection numbers in the curve graph

2014 ◽  
Vol 46 (5) ◽  
pp. 989-1002 ◽  
Author(s):  
Tarik Aougab ◽  
Samuel J. Taylor
Keyword(s):  
Intersection Numbers ◽  
Curve Graph

scholarly journals The geometry of the curve graph of a right-angled Artin group

2014 ◽  
Vol 24 (02) ◽  
pp. 121-169 ◽  
Author(s):  
Sang-Hyun Kim ◽  
Thomas Koberda
Keyword(s):  
Mapping Class Groups ◽  
Mapping Class ◽  
Artin Group ◽  
Artin Groups ◽  
Class Groups ◽  
Curve Graph ◽  
Right Angled Artin Groups ◽  
Central Result ◽  
The Right ◽  
Image Theorem

We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph, respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result, we are able to develop a Nielsen–Thurston classification for elements in the right-angled Artin group. Our analogy spans both the algebra regarding subgroups of right-angled Artin groups and mapping class groups, as well as the geometry of the extension graph and the curve graph. On the geometric side, we establish an analogue of Masur and Minsky's Bounded Geodesic Image Theorem and their distance formula.

scholarly journals An upper bound on the asymptotic translation lengths on the curve graph and fibered faces

2021 ◽  
Vol 70 (4) ◽  
pp. 1625-1637
Author(s):  
Hyungryul Baik ◽  
Hyunshik Shin ◽  
Chenxi Wu
Keyword(s):  
Upper Bound ◽  
Curve Graph

scholarly journals From Hierarchical to Relative Hyperbolicity

2020 ◽  
Author(s):  
Jacob Russell
Keyword(s):  
Hyperbolic Space ◽  
Teichmüller Space ◽  
Hyperbolic Groups ◽  
Teichmuller Space ◽  
Curve Graph ◽  
New Formulation ◽  
Relatively Hyperbolic ◽  
Relative Hyperbolicity

Abstract We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our criteria to graphs associated to surfaces and prove that the separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures. We also recover a celebrated theorem of Brock and Masur on the relative hyperbolicity of the Weil–Petersson metric on Teichmüller space for surfaces with complexity three.

scholarly journals The shadow of a Thurston geodesic to the curve graph

2015 ◽  
Vol 8 (4) ◽  
pp. 1085-1118 ◽  
Author(s):  
Anna Lenzhen ◽  
Kasra Rafi ◽  
Jing Tao
Keyword(s):  
Curve Graph

scholarly journals Origami edge-paths in the curve graph

2021 ◽  
pp. 107730
Author(s):  
Hong Chang ◽  
Xifeng Jin ◽  
William W. Menasco
Keyword(s):  
Curve Graph

scholarly journals Minimal Asymptotic Translation Lengths of Torelli Groups and Pure Braid Groups on the Curve Graph

2018 ◽  
Vol 2020 (24) ◽  
pp. 9974-9987
Author(s):  
Hyungryul Baik ◽  
Hyunshik Shin
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Braid Groups ◽  
Mapping Class ◽  
Torelli Group ◽  
Pure Braid Group ◽  
Curve Graph ◽  
Translation Length

Abstract In this paper, we show that the minimal asymptotic translation length of the Torelli group ${\mathcal{I}}_g$ of the surface $S_g$ of genus $g$ on the curve graph asymptotically behaves like $1/g$, contrary to the mapping class group ${\textrm{Mod}}(S_g)$, which behaves like $1/g^2$. We also show that the minimal asymptotic translation length of the pure braid group ${\textrm{PB}}_n$ on the curve graph asymptotically behaves like $1/n$, contrary to the braid group ${\textrm{B}}_n$, which behaves like $1/n^2$.

Rigidity of the nonseparating and outer curve graph

2018 ◽  
Vol 26 (1) ◽  
pp. 75-97
Author(s):  
Jesús Hernández Hernández
Keyword(s):  
Curve Graph ◽  
Outer Curve
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