Reducible Diagrams and Equations Over Groups

Gersten, S. M.

doi:10.1007/978-1-4613-9586-7_2

Cited by 100 publications

(123 citation statements)

References 13 publications

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“…The standard split extension of G n .k; l/ by the cyclic group of order n has presentation E n .k; l/ D hx; t j t n ; xt [6] gives that if R n .k; l/ is aspherical (in the sense that any non-empty spherical picture over R contains a dipole) then the presentation P n .k; l/ is aspherical (more precisely, it is diagrammatically reducible in the sense of Gersten [9], which implies that 2 .K/ D 0, where K is the standard CW-complex associated with P ). Theorem 4.1 of [3] gives necessary and sufficient conditions for R n .k; l/ to be aspherical.…”

Section: Asphericitymentioning

confidence: 99%

The cyclically presented groups with relators $x_ix_{i+k}x_{i+l}$

Edjvet¹,

Williams²

2010

Groups Geom. Dyn.

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Abstract. Continuing Cavicchioli, Repovš, and Spaggiari's investigations into the cyclic presentations hx 1 ; : : : ; x n j x i x i Ck x i Cl D 1 .1 Ä i Ä n/i we determine when they are aspherical and when they define finite groups; in these cases we describe the groups' structures. In many cases we show that if the group is infinite then it contains a non-abelian free subgroup. Mathematics Subject Classification (2010). 20F05, 20F06.

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Section: Asphericitymentioning

confidence: 99%

The cyclically presented groups with relators $x_ix_{i+k}x_{i+l}$

Edjvet¹,

Williams²

2010

Groups Geom. Dyn.

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“…If open cells are mapped homeomorphically to open cells, then f is called a spherical diagram over P . f is called reducible if there is a pair of 2-cells in C having a boundary edge t in common and being mapped onto the same 2-cell in K P by folding over t. A 2-complex K is called diagrammatically reducible (DR) if each spherical diagram over K is reducible (for details see [3] or [10]). …”

Section: Some Graphsmentioning

confidence: 99%

Aspherical Labelled Oriented Trees and Knots

Huck

Rosebrock

2001

Proceedings of the Edinburgh Mathematical Society

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“…Similarly, summing κ(f ) over all faces of X counts the number of faces minus the sum of all the assigned external angles. Combining these two counts yields the following theorem, versions of which have been proven by Lyndon [63], Gersten [47], Pride [75], Ballmann and Buyalo [3], and myself and Wise [67], among others. See Section 4 of [67] for a more detailed discussion of its history.…”

Section: Combinatorial Gauss-bonnetmentioning

confidence: 76%

“…The first involves the notion of conformally CAT(0) 2-complexes as developed by Gersten [47] and investigated further by Pride [75], Corson [42], Huck and Rosebrock [53] and others. The second application uses a more recent notion due to Dani Wise [84] that he calls nonpositive section curvature.…”

Section: Conformal Cat(0) and Sectional Curvaturementioning

confidence: 99%

“…Conformally CAT(0) 2-complexes have a number of nice properties. Steve Gersten has shown, for example, that conformally CAT(0) complexes are aspherical [47] and Jon Corson has shown that conformally CAT(−1) complexes (with all internal angles positive) are word-hyperbolic [42]. It is an interesting question whether either of these results can be extended to higher dimensions.…”

Section: Conformal Cat(0) and Sectional Curvaturementioning

confidence: 99%

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Constructing non-positively curved spaces and groups

McCammond¹

2009

Geometric and Cohomological Methods in Group Theory

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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

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Reducible Diagrams and Equations Over Groups

Cited by 100 publications

References 13 publications

The cyclically presented groups with relators $x_ix_{i+k}x_{i+l}$

The cyclically presented groups with relators $x_ix_{i+k}x_{i+l}$

Aspherical Labelled Oriented Trees and Knots

Constructing non-positively curved spaces and groups

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