From Wall Spaces to Cat(0) Cube Complexes

Chatterji, Indira; Niblo, Graham A.

doi:10.1142/s0218196705002669

Cited by 91 publications

(97 citation statements)

References 9 publications

(11 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These are examples of Dunwoody's "tracks" and we expect they will be referred to as "walls" in future work on this subject. Indeed, subsequently, Nica [Nic04] and Chatterji and Niblo [CN05] have written out an explicit application of Sageev's construction to cubulate abstract "spaces with walls". Those were introduced by Haglund and Paulin [HP98] especially motivated by Coxeter groups and CAT(0) cube complexes.…”

Section: Hypergraphs and Carriersmentioning

confidence: 99%

Cubulating random groups at density less than $1/6$

Ollivier

Wise

2011

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Section: Hypergraphs and Carriersmentioning

confidence: 99%

Cubulating random groups at density less than $1/6$

Ollivier

Wise

2011

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…In order to get the above statement, we prove the existence of a "space with walls" structure as introduced by Haglund and Paulin [11]. A theorem of Chatterji-Niblo [4] or Nica [14] (for similar constructions in other settings, see also [13] or [15]) gives the announced action on a CAT(0) cube complex.…”

Section: Theoremmentioning

confidence: 99%

“…By Proposition 4.2, G acts properly on (G, W). By [4] G acts properly isometrically on some I(W)-dimensional cube complex where I(W) is the supremum of the cardinalities of collections of walls which pairwise cross (see Theorem 1.4).…”

Section: Corollary 52mentioning

confidence: 99%

“…Arzhantseva, M. Lustig and Y. Stalder for their remarks about preliminary versions of the paper. Last but not least, many thanks are due to the referee, who provided a great help to correct some mistakes and gave invaluable comments to make the paper better: among many other things he indicated to the author the results of Chatterji-Niblo [4] and Nica [14], and gave him the idea of looking for the dimension of the cube complex.…”

Section: ±1mentioning

confidence: 99%

See 1 more Smart Citation

A non-trivial example of a free-by-free group with the Haagerup property

Gautero¹

2012

Groups Geom. Dyn.

View full text Add to dashboard Cite

show abstract

“…Partly due to the ease with which curvature can be checked (Lemma 7.1), these cube complexes have become a favorite of geometric group theorists [9,37,72,73,77,78]. In this section and the next I discuss how high dimensional cube complexes can be used to prove that various groups are CAT(0) groups.…”

Section: Small Cancellation Groupsmentioning

confidence: 99%

Constructing non-positively curved spaces and groups

McCammond¹

2009

Geometric and Cohomological Methods in Group Theory

View full text Add to dashboard Cite

Abstract. The theory of non-positively curved spaces and groups is tremendously powerful and has enormous consequences for the groups and spaces involved. Nevertheless, our ability to construct examples to which the theory can be applied has been severely limited by an inability to test -in real timewhether a random finite piecewise Euclidean complex is non-positively curved. In this article I focus on the question of how to construct examples of nonpositively curved spaces and groups, highlighting in particular the boundary between what is currently do-able and what is not yet feasible. Since this is intended primarily as a survey, the key ideas are merely sketched with references pointing the interested reader to the original articles.Over the past decade or so, the consequences of non-positive curvature for geometric group theorists have been throughly investigated, most prominently in the book by Bridson and Haefliger [26]. See also the recent review article by Kleiner in the Bulletin of the AMS [59] and the related books by Ballmann [4], BallmannGromov-Schroeder [5] and the original long article by Gromov [48]. In this article I focus not on the consequences of the theory, but rather on the question of how to construct examples to which it applies. The structure of the article roughly follows the structure of the lectures I gave during the Durham symposium with the four parts corresponding to the four talks. Part 1 introduces the key problems and presents some basic decidability results, Part 2 focuses on practical algorithms and low-dimensional complexes, Part 3 presents case studies involving special classes of groups such as Artin groups, small cancellation groups and ample-twisted face pairing 3-manifolds. In Part 4 I explore the weaker notion of conformal non-positive curvature and introduce the notion of an angled n-complex. As will become clear, the topics covered have a definite bias towards research in which I have played some role. These are naturally the results with which I am most familiar and I hope the reader will pardon this lack of a more impartial perspective. Part 1. Negative curvatureAlthough the definitions of δ-hyperbolic and CAT(0) spaces / groups are wellknown, it is perhaps under-appreciated that there are at least seven potentially distinct classes of groups which all have some claim to the description "negativelycurved." Sections 1 and 2 briefly review the definitions needed in order to describe these classes, the known relationships between them, and the current status of the (rather optimisitic) conjecture that all seven of these classes are identical. Since the emphasis in this article is on the construction of examples, finite piecewise

show abstract

From Wall Spaces to Cat(0) Cube Complexes

Abstract: Abstract. We explain how to adapt a construction due to M. Sageev in order to construct a proper action of a group on a CAT(0) cube complex starting from a proper action of the group on a wall space.

Cited by 91 publications

References 9 publications

Cubulating random groups at density less than $1/6$

Cubulating random groups at density less than $1/6$

A non-trivial example of a free-by-free group with the Haagerup property

Constructing non-positively curved spaces and groups

Contact Info

Product

Resources

About