Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInfinite-type loxodromic isometries of the relative arc graph
360
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
An infinite-type surface $Sigma$ is of type $mathcal{S}$ if it has an isolated puncture $p$ and admits shift maps. This includes all infinite-type surfaces with an isolated puncture outside of two sporadic classes. Given such a surface, we construct an infinite family of intrinsically infinite-type mapping classes that act loxodromically on the relative arc graph $mathcal{A}(Sigma, p)$. J. Bavard produced such an element for the plane minus a Cantor set, and our result gives the first examples of such mapping classes for all other surfaces of type $mathcal{S}$. The elements we construct are the composition of three shift maps on $Sigma$, and we give an alternate characterization of these elements as a composition of a pseudo-Anosov on a finite-type subsurface of $Sigma$ and a standard shift map. We then explicitly find their limit points on the boundary of $mathcal{A}(Sigma,p)$ and their limiting geodesic laminations. Finally, we show that these infinite-type elements can be used to prove that Map$(Sigma,p)$ has an infinite-dimensional space of quasimorphisms.
rate research
Read More
Let $X_{0}$ be a complete hyperbolic surface of infinite type with geodesic boundary which admits a countable pair of pants decomposition. As an application of the Basmajian identity for complete bordered hyperbolic surfaces of infinite type with limit sets of 1-dimensional measure zero, we define an asymmetric metric (which is called arc metric) on the quasiconformal Teichmuller space $mathcal{T}(X_{0})$ provided that $X_{0}$ satisfies a geometric condition. Furthermore, we construct several examples of hyperbolic surfaces of infinite type satisfying the geometric condition and discuss the relation between the Shigas condition and the geometric condition.
In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These {it multiarc graphs} naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and topology. We show a number of rigidity results, namely showing that, under certain complexity conditions, that simplicial maps between them only arise in the obvious way. We also observe that, again under necessary complexity conditions, subsurface strata are convex. Put together, these results imply that certain simplicial maps always give rise to convex images.
We associate to triangulations of infinite type surface a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of our results is that flip graphs for infinite type surfaces have uncountably many connected components.
Let ${bf H}_{mathbb C}^n$ be the $n$-dimensional complex hyperbolic space and ${rm SU}(n,1)$ be the (holomorphic) isometry group. An element $g$ in ${rm SU}(n,1)$ is called loxodromic or hyperbolic if it has exactly two fixed points on the boundary $partial {bf H}_{mathbb C}^n$. We classify ${rm SU}(n,1)$ conjugation orbits of pairs of loxodromic elements in ${rm SU}(n,1)$.
We show that any surface of infinite type admits an ideal triangulation. Furthermore, we show that a set of disjoint arcs can be completed into a triangulation if and only if, as a set, they intersect every simple closed curve a finite number of times.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
