scholarly journals Uniform local finiteness of the curve graph via subsurface projections

2016 ◽  
Vol 10 (4) ◽  
pp. 1265-1286
Author(s):  
Yohsuke Watanabe
Keyword(s):  
Local Finiteness ◽  
Curve Graph

scholarly journals The local structure of the free boundary in the fractional obstacle problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Matteo Focardi ◽  
Emanuele Spadaro
Keyword(s):  
Hausdorff Dimension ◽  
Free Boundary ◽  
Hausdorff Measure ◽  
Local Structure ◽  
Obstacle Problem ◽  
List Type ◽  
Minkowski Content ◽  
Local Finiteness ◽  
Lower Dimensional

AbstractBuilding upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results:(i)local finiteness of the {(n-1)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\mathcal{H}}^{n-1}}-rectifiability of the free boundary,(iii)classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most {(n-2)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {k\geq 2}, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than {k+1}.

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 On the laws of certain finite groups

1970 ◽  
Vol 11 (4) ◽  
pp. 441-489 ◽  
Author(s):  
John Cossey ◽  
Sheila Oates MacDonald ◽  
Anne Penfold Street
Keyword(s):  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Classification Problems ◽  
Sylow Subgroups ◽  
Local Finiteness ◽  
Special Cases

In recent years a great deal of attention has been devoted to the study of finite simple groups, but one aspect which seems to have been little considered is that of the laws they satisfy. In a recent paper [3], the first two of the present authors gave a basis for laws of PSL(2, 5). The techniques of [3] can be used to show that (modulo certain classification problems) a basis for the laws of PSL(2, pn) can be made up from laws of the following types:(1) an exponent law,(2) laws which determine the Sylow subgroups,(3) laws which determine the normalisers of the Sylow subgroups,(4) in certain special cases, laws which determine subvarieties of smaller exponent, e.g. the subvariety of exponent 12 for those PSL(2, pn) which contain S4,(5) a law implying local finiteness.

scholarly journals Rigidity of the strongly separating curve graph

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

Some groups of exponent 72

2014 ◽  
Vol 17 (6) ◽  
Author(s):  
Enrico Jabara ◽  
Daria V. Lytkina ◽  
Victor D. Mazurov
Keyword(s):  
Local Finiteness

AbstractLocal finiteness is proved for groups of exponent dividing 72 with no elements of order 6.

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
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