Systolic invariants of groups and 2-complexes via Grushko decomposition

Rudyak, Yuli B.; Sabourau, Stéphane

doi:10.5802/aif.2369

Cited by 17 publications

(25 citation statements)

References 25 publications

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“…This bound was proved by Rudyak & Sabourau in [RS08] where the authors also posed the following two fundamental questions: given a non-free group G, is it true that (1) σ(G) ≥ 2/π ?…”

Section: Introductionmentioning

confidence: 96%

Systolic geometry and simplicial complexity for groups

Babenko¹,

Balacheff

Bulteau³

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Twenty years ago Gromov asked about how large is the set of isomorphism classes of groups whose systolic area is bounded from above. This article introduces a new combinatorial invariant for finitely presentable groups called {\it simplicial complexity} that allows to obtain a quite satisfactory answer to his question. Using this new complexity, we also derive new results on systolic area for groups that specify its topological behaviour.Comment: 35 pages, 9 figure

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“…This bound was proved by Rudyak & Sabourau in [RS08] where the authors also posed the following two fundamental questions: given a non-free group G, is it true that (1) σ(G) ≥ 2/π ?…”

Section: Introductionmentioning

confidence: 96%

Systolic geometry and simplicial complexity for groups

Babenko¹,

Balacheff

Bulteau³

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Note that the finite-dimensional approximation is used in an analytic proof of Gromov's systolic inequality by Ambrosio and the second-named author in [1]. Recent publications in systolic geometry include the articles [2][3][4][5][6][7][8][9]13,15,16,18,19,21,22,27,28].…”

Section: Theorem 11 Let M Be a Compact Riemannian Manifold Without Bmentioning

confidence: 99%

Bi-Lipschitz approximation by finite-dimensional imbeddings

Katz

2010

Geom Dedicata

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Gromov's celebrated systolic inequality from '83 is a universal volume lower bound in terms of the least length of a noncontractible loop in M. His proof passes via a strongly isometric imbedding called the Kuratowski imbedding, into the Banach space of bounded functions on M. We show that the imbedding admits an approximation by a (1 + )-bi-Lipschitz (onto its image), finite-dimensional imbedding for every > 0, using the first variation formula and the mean value theorem. Mathematics Subject Classification (2000)Primary 53C23 · Secondary 26E35 Metric imbeddings and Gromov's theoremIn '83, Gromov proved the systolic inequality for essential manifolds M. Namely, consider the least length (systole, denoted "sys") of a non-contractible loop in a closed Riemannian manifold M. Then the systole is bounded above in terms of the volume of M, if M satisfies the topological hypothesis of being essential (for instance, if π i (M) = 0 for i ≥ 2). See Guth [14] for a helpful overview.A key technique in Gromov's seminal text [11] is the Kuratowski imbedding. Thus, Gromov imbeds a Riemannian manifold M into the space

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“…Recent developments in systolic geometry include [1][2][3][4][5][7][8][9][10][11][12]15,16,18,17,20,21,23,24,26,28,29,[31][32][33][34]38]. …”

Section: Congruence Towers and The 4/3 Boundmentioning

confidence: 99%

Hurwitz quaternion order and arithmetic Riemann surfaces

2011

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We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry. Mathematics Subject Classification (2000)11R52 · 53C23 · 16K20 Congruence towers and the 4/3 boundHurwitz surfaces are an important and famous family of Riemann surfaces.We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry. 1 A Hurwitz surface X by definition attains the upper bound of 84(g X − 1) for the order |Aut(X )| of the holomorphic automorphism group of X , where g X is its genus. In [25], we proved a systolic boundIn the literature, the term "Hurwitz quaternion order" has been used both in the sense used in the present text, and in the sense of the unique maximal order of Hamilton's rational quaternions (−1, −1) Q .

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Systolic invariants of groups and 2-complexes via Grushko decomposition

Cited by 17 publications

References 25 publications

Systolic geometry and simplicial complexity for groups

Systolic geometry and simplicial complexity for groups

Bi-Lipschitz approximation by finite-dimensional imbeddings

Hurwitz quaternion order and arithmetic Riemann surfaces

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