An introduction to asymptotic geometry

2012 ◽  
pp. 405-454 ◽  
Author(s):  
Viktor Schroeder
Keyword(s):  
Asymptotic Geometry

scholarly journals Coarse median algebras: the intrinsic geometry of coarse median spaces and their intervals

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Graham A. Niblo ◽  
Nick Wright ◽  
Jiawen Zhang
Keyword(s):  
Finite Rank ◽  
Intrinsic Geometry ◽  
Bounded Geometry ◽  
Higher Dimensional ◽  
Median Algebra ◽  
Asymptotic Geometry ◽  
Median Operator

AbstractThis paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of $$\delta $$ δ -hyperbolicity.

scholarly journals The asymptotic geometry of right-angled Artin groups, I

2008 ◽  
Vol 12 (3) ◽  
pp. 1653-1699 ◽  
Author(s):  
Mladen Bestvina ◽  
Bruce Kleiner ◽  
Michah Sageev
Keyword(s):  
Artin Groups ◽  
Right Angled Artin Groups ◽  
Asymptotic Geometry

scholarly journals Law of large numbers for the drift of the two-dimensional wreath product

2021 ◽  
Author(s):  
Anna Erschler ◽  
Tianyi Zheng
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Wreath Product ◽  
Law Of Large Numbers ◽  
Abelian Groups ◽  
Limiting Distribution ◽  
Two Dimensional ◽  
Lamplighter Group ◽  
Large Numbers ◽  
Asymptotic Geometry

AbstractWe prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $$(2+\epsilon )$$ ( 2 + ϵ ) -moment. This result is in contrast with classical examples of abelian groups, where the displacement after n steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised n-step displacement admits a density whose support is $$[0,\infty )$$ [ 0 , ∞ ) . We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not hold, and study relation of this property with asymptotic geometry of groups.

scholarly journals Fine asymptotic geometry in the Heisenberg group

2014 ◽  
Vol 63 (3) ◽  
pp. 885-916 ◽  
Author(s):  
Moon Duchin ◽  
Christoper Mooney
Keyword(s):  
Heisenberg Group ◽  
Asymptotic Geometry

scholarly journals Asymptotic Geometry of Discrete Interlaced Patterns: Part II

2020 ◽  
Vol 70 (1) ◽  
pp. 375-436
Author(s):  
Erik Duse ◽  
Anthony Metcalfe
Keyword(s):  
Asymptotic Geometry

scholarly journals LOCAL SET APPROXIMATION: MATTILA–VUORINEN TYPE SETS, REIFENBERG TYPE SETS, AND TANGENT SETS

2015 ◽  
Vol 3 ◽  
Author(s):  
MATTHEW BADGER ◽  
STEPHEN LEWIS
Keyword(s):  
Measure Theory ◽  
Geometric Measure Theory ◽  
Singular Part ◽  
Variational Problems ◽  
Geometric Measure ◽  
Model Sets ◽  
Tangent Sets ◽  
Asymptotic Geometry ◽  
Set Approximation ◽  
Type Sets

We investigate the interplay between the local and asymptotic geometry of a set $A\subseteq \mathbb{R}^{n}$ and the geometry of model sets ${\mathcal{S}}\subset {\mathcal{P}}(\mathbb{R}^{n})$, which approximate $A$ locally uniformly on small scales. The framework for local set approximation developed in this paper unifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, and Preiss. We indicate several applications of this framework to variational problems that arise in geometric measure theory and partial differential equations. For instance, we show that the singular part of the support of an $(n-1)$-dimensional asymptotically optimally doubling measure in $\mathbb{R}^{n}$ ($n\geqslant 4$) has upper Minkowski dimension at most $n-4$.

scholarly journals The asymptotic geometry of teichmuller space

Topology ◽  
1980 ◽  
Vol 19 (1) ◽  
pp. 23-41 ◽  
Author(s):  
Steven P. Kerckhoff
Keyword(s):  
Teichmüller Space ◽  
Teichmuller Space ◽  
Asymptotic Geometry
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