Some Duality Conjectures for Finite Graphs and Their Group Theoretic Consequences

Gersten, S. M.; Riley, Tim

doi:10.1017/s0013091503000890

Cited by 7 publications

(18 citation statements)

References 15 publications

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“…We conjecture that Questions 5.5(iii) and 5.5(iv) are, in fact, equivalent on account of a close relationship between FL and GL that is discussed further in § 7 and in [11].…”

Section: It Follows That Gl(n) a 1+2 Diam(n) For All Nmentioning

confidence: 79%

“…The second part is a consequence of Theorem 7.1 of [11], which says that for fat finite presentations, GL is -equivalent to another filling function called DlogA. The first part of this theorem follows from Proposition 5.2.…”

Section: It Follows That Gl(n) a 1+2 Diam(n) For All Nmentioning

confidence: 94%

“…In [11] we discuss a number of interrelated conjectures about complementary pairs of trees in finite planar graphs. Conjecture 4.3 claims that GL and DGL are essentially the same diagram measurement when one is concerned with diagrams for which there is a uniform bound on the length of the boundary cycles of each 2-cell (with the possible exception of the 2-cell at infinity).…”

Section: The Filling Length Of a Diagrammentioning

confidence: 99%

“…However, z will play the following important role in the proof of Theorem 4.5 below and will also be used in § 6 and in [11]. However, z will play the following important role in the proof of Theorem 4.5 below and will also be used in § 6 and in [11].…”

Section: Changing the Presentationmentioning

confidence: 99%

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The Gallery Length Filling Function and a Geometric Inequality for Filling Length

Gersten

Riley

2006

Proc. Lond. Math. Soc.

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A diagram is a finite planar 2-complex homeomorphic to a singular 2-disc. A van Kampen diagram is a labelled diagram amounting to a graphical demonstration of how a word in a finite presentation P of a group Γ that represents the identity, is a consequence of the defining relations. (See Definitions 2.1 and 4.1.)The use of diagrams to probe the geometry of finitely presented groups and to produce invariants (filling functions) is well established in Geometric Group Theory. To date, attention has focused on the area of diagrams, and the resulting Dehn function (also known as the minimal isoperimetric function), and on the diameter of the 1-skeleton of diagrams (in the combinatorial metric), which gives the minimal isodiametric function. And recently the filling length of a diagram D has proved important [9, 10]; we define FL(D) to be the minimal length L such that there is a combinatorial null-homotopy of ∂D, across D and down to a base vertex, through loops of length at most L. (See § 2.1 for more details.)The innovation in this article is to bring duality considerations to bear on the study of diagrams. We define the gallery length GL(D) of a diagram D to be the diameter of the 1-skeleton of its dual. We will also define DGL(D) which combines diameter and gallery length as realisable on complementary pairs of maximal trees; see § 2.3. In § 2 we give formal definitions of diagrams and of GL(D) and DGL(D) as well as other diagram measurements, and in Proposition 2.4 we list some of the inequalities that relate diagram measurements. Then in § 3 we use DGL to control the filling length of a diagram.Theorem 3.5. Fix λ > 0. There exists K > 0 such that for every diagram D in which the boundary of each 2-cell has at most λ edges, FL(D) K(DGL(D) + Perimeter(D)).This inequality is an amendment of a suggestion of Gromov [13, § 5.C]: the term DGL(D) replaces Diam(D). Gromov's inequality was known to fail [6] in the context of 2-discs with Riemannian metrics. In § 3 we give our own family of diagrams D n that have bounded perimeter and diameter, and at most three edges in the boundary of each 2-cell, but have filling length tending to infinity. It follows that Gromov's inequality does not hold in a combinatorial context. Our proof that FL(D n ) → ∞ takes us into unlikely territory. We use the following result, proved in an appendix, concerning the number n of terms required to express an integer n as a sum of Mathematics Subject Classification 20F05 (primary), 20F06, 57M05, 57M20 (secondary). s. m. gersten and t. r. riley which GL Q (D w ) = GL Q (w). The relations in Q that involve t are tu −1 and [t ±1 , z]. It follows that all occurrences of t in D w must be in 't-corridors', that is, concatenations of 2-cells with boundary labels [t, z] ±1 and with adjacent 2-cells joined across an edge labelled t. There are two types of t-corridor: linear t-corridors start with a 2-cell labelled by (tu −1 ) ±1 and finish with a 2-cell labelled (tu −1 ) ±1

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“…We conjecture that Questions 5.5(iii) and 5.5(iv) are, in fact, equivalent on account of a close relationship between FL and GL that is discussed further in § 7 and in [11].…”

Section: It Follows That Gl(n) a 1+2 Diam(n) For All Nmentioning

confidence: 79%

Section: It Follows That Gl(n) a 1+2 Diam(n) For All Nmentioning

confidence: 94%

Section: The Filling Length Of a Diagrammentioning

confidence: 99%

Section: Changing the Presentationmentioning

confidence: 99%

See 2 more Smart Citations

The Gallery Length Filling Function and a Geometric Inequality for Filling Length

Gersten

Riley

2006

Proc. Lond. Math. Soc.

Self Cite

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show abstract

“…It was open for so long because all known methods to construct algorithmically hard groups produced either non-residually finite groups or groups where the question about their residual finiteness is very difficult. Not much is known even for linear groups (note that Gersten asked [19,20] if there exists a uniform upper bound for Dehn functions of linear groups). Let us briefly discuss the previous attempts to solve the problem and the reasons why these methods did not work.…”

Section: The Problem and Previous Approaches For A Solutionmentioning

confidence: 99%