Conjugation curvature for Cayley graphs

Bar-Natan, Assaf; Duchin, Moon; Kropholler, Robert

doi:10.1142/s1793525321500096

Cited by 6 publications

(9 citation statements)

References 11 publications

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“…This must also be a minimizer within F as ∆ and F are invariant under adding constants. This proves (1).…”

Section: Extremal Lipschitz Extensionsupporting

confidence: 53%

“…Proof. We first prove (1). By Lemma 3.3, there exists f ∈ F with ∆f = c = const on K, and by Lemma 3.4, we have c = 0 proving (1).…”

Section: Existence Of Lipschitz Harmonic Functionsmentioning

confidence: 84%

“…x n = y) be a path from x to y in G . We notice x n , x n−1 ∈ K 1 as f (x n ) < a + 1 and f (x n−1 ) < a + 2 ≤ b as f ∈ Lip (1). By the same argument, the path has to contain at least two vertices in K 2 as the only way to exit M a within G is through K 2 .…”

Section: Connected Level Setsmentioning

confidence: 96%

“…In the last few years, several authors tried to establish a discrete Cheeger-Gromoll theorem, usually without success due to the potential counterexamples. The best results so far in this direction are splitting theorems for groups and their Cayley graphs [1,35]. However, it is still open whether a splitting theorem holds for general graphs.…”

Section: History Of Proof Attemptsmentioning

confidence: 99%

“…As the curvature at the corner equals the vertex degree, we see that κ ≥ 0 also after gluing. Moreover, m(n, n) = 1 2 for all n ∈ Z \ {0} showing that both ends have infinite measure. Hence, G is a salami.…”

Section: Why Edge Weight Bounded From Below?mentioning

confidence: 99%

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Every Salami has two ends

Hua¹,

Münch²

2021

Preprint

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A salami is a connected, locally finite, weighted graph with non-negative Ollivier Ricci curvature and at least two ends of infinite volume. We show that every salami has exactly two ends and no vertices with positive curvature. We moreover show that every salami is recurrent and admits harmonic functions with constant gradient. The proofs are based on extremal Lipschitz extensions, a variational principle and the study of harmonic functions. Assuming a lower bound on the edge weight, we prove that salamis are quasi-isometric to the line, that the space of all harmonic functions has finite dimension, and that the space of subexponentially growing harmonic functions is two-dimensional. Moreover, we give a Cheng-Yau gradient estimate for harmonic functions on balls.

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“…This must also be a minimizer within F as ∆ and F are invariant under adding constants. This proves (1).…”

Section: Extremal Lipschitz Extensionsupporting

confidence: 53%

“…Proof. We first prove (1). By Lemma 3.3, there exists f ∈ F with ∆f = c = const on K, and by Lemma 3.4, we have c = 0 proving (1).…”

Section: Existence Of Lipschitz Harmonic Functionsmentioning

confidence: 84%

Section: Connected Level Setsmentioning

confidence: 96%

Section: History Of Proof Attemptsmentioning

confidence: 99%

Section: Why Edge Weight Bounded From Below?mentioning

confidence: 99%

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Every Salami has two ends

Hua¹,

Münch²

2021

Preprint

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A new proof of the growth rate of the solvable Baumslag–Solitar groups

Taback

Walker

2022

Geom Dedicata

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We exhibit a regular language of geodesics for a large set of elements of BS(1, n) and show that the growth rate of this language is the growth rate of the group. This provides a straightforward calculation of the growth rate of BS(1, n), which was initially computed by Collins, Edjvet and Gill in [5]. Our methods are based on those we develop in [8] to show that BS(1, n) has a positive density of elements of positive, negative and zero conjugation curvature, as introduced by Bar-Natan, Duchin and Kropholler in [1].

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Dead ends on wreath products and lamplighter groups

Silva

2023

Int. J. Algebra Comput.

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For any finite group [Formula: see text] and any finitely generated group [Formula: see text], we prove that the corresponding lamplighter group [Formula: see text] admits a standard generating set with unbounded depth, and that if [Formula: see text] is abelian then the above is true for every standard generating set. This generalizes the case where [Formula: see text] together with its cyclic generator [S. Cleary and J. Taback, Dead end words in lamplighter groups and other wreath products, Q. J. Math. 56(2) (2005) 165–178]. When [Formula: see text] is the free product of two finite groups [Formula: see text] and [Formula: see text], we characterize which standard generators of the associated lamplighter group have unbounded depth in terms of a geometrical constant related to the Cayley graphs of [Formula: see text] and [Formula: see text]. In particular, we find differences with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups has uniformly bounded depth with respect to some standard generating set.

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Conjugation curvature for Cayley graphs

Cited by 6 publications

References 11 publications

Every Salami has two ends

Every Salami has two ends

A new proof of the growth rate of the solvable Baumslag–Solitar groups

Dead ends on wreath products and lamplighter groups

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