Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0704.1922

Relative Rigidity, Quasiconvexity and C-Complexes

Mahan Mj
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce and study the notion of relative rigidity for pairs $(X,\JJ)$ where 1) $X$ is a hyperbolic metric space and $\JJ$ a collection of quasiconvex sets 2) $X$ is a relatively hyperbolic group and $\JJ$ the collection of parabolics 3) $X$ is a higher rank symmetric space and $\JJ$ an equivariant collection of maximal flats Relative rigidity can roughly be described as upgrading a uniformly proper map between two such $\JJ$'s to a quasi-isometry between the corresponding $X$'s. A related notion is that of a $C$-complex which is the adaptation of a Tits complex to this context. We prove the relative rigidity of the collection of pairs $(X, \JJ)$ as above. This generalises a result of Schwarz for symmetric patterns of geodesics in hyperbolic space. We show that a uniformly proper map induces an isomorphism of the corresponding $C$-complexes. We also give a couple of characterizations of quasiconvexity. of subgroups of hyperbolic groups on the way.

Explore related subjects

math.GTmath.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Mahan Mj. 2008-08-04. Relative Rigidity, Quasiconvexity and C-Complexes. https://doi.org/10.2140/agt.2008.8.1691

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Word Length Formulae, Normal Forms, Conjugation and Root-finding Algorithms in Surface Groups

In this paper, we mainly study the following symmetric presentation of the surface group $$π_1(Σ_g)=\left\langle c_1,\dots, c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\right\rangle.$$ For every nontrivial element $x\in π_1(Σ_g)$ and $k\geq 2$, we obtain a uniform representative of the normal forms $\mathfrak{nf}(x^k)$ of $x^k$ under the length-lexicographical order: $$\mathfrak{nf}(x^k) = \overline{LW^{k-2}R}.$$ Building on this result, we establish a new relation among these normal forms, and then derive the following three formulae related to the word length: $|x^2|>|x|$; $|x^k|=(k-1)(|x^2|-|x|)+|x|$; $\lim_{k\to\infty}\frac{|x^k|}{k}=|x^2|-|x|$. Furthermore, we extend these results to obtain a coarser analogue for every minimal geometric presentation. We then define normal forms of conjugacy classes in $π_1(Σ_g)$ and provide a criterion for determining the conjugacy of group elements. As a consequence, we provide efficient algorithms for solving the root-finding and conjugacy problems. Finally, we present applications to the computation of several growth rates.

math.GT↗
arXiv2026

Plane separating continua inscribe rectangles

We prove the following: If $X$ is a plane separating continuum, then every embedding of $X$ into $\mathbb{R}^2$ contains the vertices of a Euclidean rectangle. We arrive to this result by extending a known result by H. Vaughan for Jordan curves to a wider class of topological objects via shape theory and Steenrod homology.

math.GT↗
arXiv2026

Every Link Has Infinitely Many Explicit Generalised T-Link Presentations

Generalised $T$-links provide a simple description of all links in $S^3$ as closures of products of standard twisting blocks, parametrised by finite sequences of integers. We prove that every link admits infinitely many pairwise distinct generalised $T$-link presentations. Starting from any such presentation, we give explicit parameter transformations that preserve the represented link and generate families of pairwise distinct presentations depending on arbitrarily many independent integer parameters.

math.GT↗