Pith. sign in

REVIEW 2 major objections 4 minor 48 references

Isoperimetric inequalities in Hadamard spaces of asymptotic rank two

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Rank-two Hadamard spaces admit near-linear isoperimetric fillings for all cycles.

desk verdict Genuinely new homological δ-isoperimetric theorem for rank-2 CAT(0) spaces, but the proof rests on a delegated density lemma that should be stated and checked in full. read the letter →

arxiv 2506.04882 v3 pith:CT6KMUS7 submitted 2025-06-05 math.MG math.DGmath.GR

Urs Lang , Stephan Stadler , David Urech This is my paper · ORCID
classification math.MGmath.DGmath.GR MSC 53C2349Q15
keywords isoperimetricgapconjectureHadamardspacesCAT(0)asymptoticrankNagatadimensionintegralcurrentsminimizingsimpliceshigherhyperbolicity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to establish Gromov's isoperimetric gap conjecture in the first nontrivial case, asymptotic rank two, and in full generality for cycles of dimension at least two. The theorem asserts that in any CAT(0) space of asymptotic rank at most two and finite asymptotic Nagata dimension, every integral k-cycle T has a filling V whose mass is at most a constant times M(T)^(1+δ), for any δ > 0. A sympathetic reader would care because this is the homological version of a phenomenon previously known only for Lipschitz 2-spheres, and because it covers concrete spaces such as Hadamard 3-manifolds and finite-dimensional CAT(0) cube complexes. The proof works by showing that minimizing simplices are uniformly slim and by decomposing arbitrary cycles into such simplices at controlled scales.

What carries the argument

The central objects are minimizing simplices: currents built inductively by filling each geodesic simplex skeleton with a mass-minimizing integral current. The key mechanism is that when the asymptotic rank is at most two, every minimizing k-simplex with k > 2 is slim: its mass is bounded by a constant times the mass of its boundary, its support lies in a bounded neighborhood of the boundary, and its facets lie close to one another (Theorem 4.4). Slimness is obtained from a uniform density estimate for minimizing 2-simplices (Lemma 4.2) propagated to higher dimensions via Proposition 4.3. The rest of the proof approximates an arbitrary cycle by piecewise minimizing cycles at multiple scales (Theorem 5.3), using finite asymptotic Nagata dimension to build controlled simplicial coverings, then fills each piece by coning and adds the small remainders by cone fillings.

What would settle it

Exhibit a CAT(0) space of asymptotic rank at most two with finite asymptotic Nagata dimension and a 2-cycle T such that every integral filling V has M(V) ≥ c M(T)^(1+δ) for some δ > 0; more directly, find a minimizing triangle whose mass inside some ball of radius r exceeds (3π/2) $r^{2}$, or a non-proper CAT(0) space where the density bound of Lemma 4.2 fails.

Watch

Extended reading notes

Core claim

The paper proves that the isoperimetric gap predicted by Gromov holds homologically in asymptotic rank two. For every CAT(0) space X of asymptotic rank at most two and finite asymptotic Nagata dimension, every integral k-cycle T with k ≥ 2 admits an integral filling V with M(V) ≤ C M(T)^(1+δ) for any δ > 0, where C depends only on X, k, and δ. The space need not be proper. This upgrades the homotopical δ-isoperimetric inequality for Lipschitz 2-spheres to a statement about arbitrary integral cycles, and it applies to all Hadamard 3-manifolds and finite-dimensional CAT(0) cube complexes.

Load-bearing premise

The whole construction depends on a uniform bound on how much mass a smallest-area triangle can pack into a ball of radius r in any CAT(0) space; if that bound fails, the control of higher-dimensional minimizing simplices collapses.

Editorial extensions

If this is right

  • Prior work gave a homotopical δ-isoperimetric inequality for Lipschitz 2-spheres; this theorem replaces 'homotopical' by 'homological' and '2-spheres' by arbitrary k-cycles for k ≥ 2.
  • For Hadamard 3-manifolds the theorem yields a near-linear isoperimetric inequality for bounded domains of finite perimeter, completing the picture up to the open question whether the exponent can be reduced to 1.
  • For finite-dimensional CAT(0) cube complexes the inequality transfers to cellular chains via the deformation theorem of geometric measure theory.
  • The argument applies uniformly to all k ≥ 2 and does not rely on the known reduction from dimension 2 to higher dimensions.
  • It remains open whether the exponent 1+δ can be improved to 1, and whether the finite asymptotic Nagata dimension hypothesis can be dropped.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The uniform density bound for minimizing 2-simplices is the real constraint: any attempt to reach asymptotic rank higher than two would need a k-dimensional analogue of Lemma 4.2, which the paper explicitly says is unclear.
  • The same approximation scheme might work with a much weaker covering assumption on the support of a single cycle rather than the whole space, so the theorem could plausibly extend to arbitrary CAT(0) spaces of asymptotic rank two without any global dimension hypothesis.
  • In concrete examples like CAT(0) cube complexes one could test sharpness by constructing explicit cycles whose filling mass grows like M^(1+δ); a linear inequality in those models would suggest the δ here is an artifact of the method.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proves a homological δ-isoperimetric filling inequality in CAT(0) spaces of asymptotic rank at most 2 and finite asymptotic Nagata dimension: for every integral k-cycle T with k ≥ 2 and every δ > 0 there is an integral filling V with ∂V = T and M(V) ≤ C M(T)^{1+δ}, where C depends only on X, k, and δ. The proof combines a theory of minimizing k-simplices (Sections 4), an approximation of arbitrary cycles by piecewise minimizing cycles controlled at multiple scales (Section 5), and an iterative filling argument using the Euclidean isoperimetric inequality and cone fillings (Section 6). The paper also records applications to Hadamard 3-manifolds and finite-dimensional CAT(0) cube complexes, and it openly identifies the restriction to asymptotic rank 2 as coming from a single density bound for minimizing 2-simplices.

Significance. If correct, this is substantial progress on Gromov's isoperimetric gap conjecture: it extends the homotopical result of Druţu–Lang–Papasoglu–Stadler for Lipschitz 2-spheres to all integral cycles of dimension at least 2, in non-proper CAT(0) spaces, under finite asymptotic Nagata dimension. The paper is honest about its hypotheses and its reliance on prior work, and no circularity is apparent; in particular, the higher-rank transfer result [45, Theorem 1.6] is not used for the main theorem. The proof structure is transparent, and the main technical novelty, the multi-scale piecewise minimizing approximation, is clearly presented. The chief reservation is that the key density estimate for minimizing 2-simplices is delegated to a thesis reference and its adaptation from compactly supported to locally integral currents is not written out.

major comments (2)
  1. [§4, Lemma 4.2] Lemma 4.2 is the unique reason for the restriction to asymptotic rank two, but its proof is only sketched: the text says that after gluing three Euclidean half-planes one obtains a locally minimizing locally integral 2-current V, and that the monotonicity formula, Corollary 3.3, applies. However, Corollary 3.3 is stated for V ∈ I_{k+1,c}(X), i.e. for compactly supported currents, whereas the constructed V is noncompact and has infinite mass; the density bound 3π/2 comes from the asymptotic behavior of the three half-planes at large radii. The paper neither states nor proves a monotonicity or density statement for locally minimizing locally integral currents in possibly non-proper CAT(0) spaces. Since Proposition 4.3, Theorem 4.4, Corollary 4.5, and Proposition 4.6 all rely on Lemma 4.2, this gap is load-bearing. Please either include a complete proof or state the precise lemma from [19, Lemma 8.4] with all hypotheses and verify that it applies to the non-proper setting used here.
  2. [§4, Proposition 4.3 and Theorem 4.4] Both results are transferred from the proper case by saying that the arguments of [20, Proposition 5.2] and [20, Theorem 1.1] apply verbatim because Theorem 3.5 and Proposition 3.2 guarantee existence of minimizing fillings and cone inequalities. This is not automatic: [20] assumes properness not only for existence of minimizers, but also for compactness and monotonicity arguments, and the paper does not identify which parts of those proofs need no local compactness. In particular, the implications (AR_{k−1}) ⇒ (LII_{k−1}), (FR_{k−1}), and (ML_{k−1}) in Theorem 4.4 are nontrivial and are stated in [20] for proper metric spaces satisfying coning inequalities. The authors should spell out the non-proper adaptation or give a precise reference to a version valid for all CAT(0) spaces.
minor comments (4)
  1. [Title and Theorem 1.1] There are two apparent typos: 'SP ACES' in the title should be 'SPACES', and 'asymptotic rank and most 2' should be 'asymptotic rank at most 2'.
  2. [§2, Lemma 2.1] In the proof of Lemma 2.1, the line 'there are always at most n + 1 non-zero summands' is correct, but the inequality displayed immediately after defining ψ could be clearer; the constant in the Lipschitz estimate for ψ is not explicitly collected at the end of the estimate.
  3. [§4, Lemma 4.1] The proof says that for k = 2 one can take c2 = 1/2 by Proposition 3.2; this is plausible because a minimizing filling of a geodesic triangle in a CAT(0) space is no larger than the geodesic cone from one vertex over the opposite side, but a one-sentence justification would help the reader.
  4. [§5, Proposition 5.1] The notation T_j and U_j is introduced before the displayed claim; it may be worth adding a sentence that the subcurrents are chosen successively with slices, as in the classical slicing argument, to make the construction easier to follow.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity found: the rank-2 delta-isoperimetric gap is derived from independent density and slimness results; the paper's self-citations are legitimate support, not restatements of the target.

full rationale

The main theorem (Theorem 1.1/6.2) is derived through a genuine chain. The load-bearing step is Lemma 4.2, the uniform density bound (3pi/2)r^2 for minimizing 2-simplices, which the paper explicitly delegates: 'It can be shown that the minimizing simplex together with the three added half-planes ... forms a (locally) minimizing locally integral 2-current V in X''. Now the result follows from the monotonicity formula, Corollary 3.3, applied to V. See [19, Lemma 8.4] for the details.' This lemma is an external, published, parameter-free result (Goldhirsch's thesis), not a restatement of the target inequality; the extension trick is credited to [41] (Stadler), an independent JEMS paper. Proposition 4.3 and Theorem 4.4 (controlled density, slimness, linear isoperimetric bound for minimizing k-simplices) import the implications (AR_{k-1}) => (LII_{k-1}), (FR_{k-1}), (ML_{k-1}) from [20, Theorem 1.1] and [30, Theorem 5.1]; although these works include the present authors, they are independent theorems whose assumptions (asymptotic rank) do not include the conclusion, and the paper notes the non-proper extension is justified by Theorem 3.5 and Proposition 3.2. Proposition 4.6 then yields conical fillings of piecewise minimizing cycles. The genuinely new part (Section 5, Theorem 5.3) approximates an arbitrary cycle by a piecewise minimizing one, independent of the rank, and Section 6 iterates it with Wenger's decomposition and external fillings (Theorems 3.4, 3.2, Proposition 6.1). At no point is the delta-isoperimetric inequality assumed; the rank-2 hypothesis enters only through the density bound, and the paper states this honestly: 'The validity of an analogous bound for higher-dimensional simplices is unclear, and this is the only reason for the restriction to asymptotic rank 2 in Theorem 1.1.' The technical caveat - Corollary 3.3 is stated for compactly supported minimizers while Lemma 4.2's V is a non-compact locally integral current - is a delegation/correctness risk, not circularity. There are no fitted parameters. Self-citations ([20], [30], [41]) are load-bearing but legitimate independent support, so the score stays low.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The theorem is conditional on standard results in metric current theory and on two less standard ingredients: the uniform density bound for minimizing 2-simplices and the extension of higher-rank hyperbolicity characterizations to non-proper spaces. No free parameters are fitted to data and no new entities are introduced; the constants are existential and depend only on X, k, and δ.

assumptions (7)
  • domain assumption X is a complete geodesic CAT(0) space of asymptotic rank at most 2.
    The ambient class of Theorem 1.1; all fillings and simplices are constructed in this setting.
  • domain assumption X has finite asymptotic Nagata dimension, also called finite linearly controlled asymptotic dimension.
    Needed for the controlled coverings in Lemma 2.1 and the approximation result Theorem 5.3.
  • standard math Existence of minimizing fillings for every integral cycle in a CAT(0) space (Theorem 3.5).
    From [42, Theorem 1.6]; used to define minimizing simplices in Section 4.
  • standard math Euclidean isoperimetric inequality and coning inequality for metric currents in CAT(0) spaces.
    Proposition 3.2 and Theorem 3.4 from [43] and [42] are used throughout for mass bounds.
  • standard math Uniform density bound for minimizing 2-simplices in CAT(0) spaces (Lemma 4.2).
    The only reason for the rank-2 restriction; the proof is delegated to [19, Lemma 8.4] and not reproduced.
  • domain assumption Results of [20] and [30] apply verbatim to non-proper CAT(0) spaces (Proposition 4.3 and Theorem 4.4).
    The paper asserts the extension without proof; the cited references assumed properness.
  • standard math Deformation lemma for metric currents in Euclidean simplicial complexes (Lemma 5.2).
    Taken from [7, Proposition A.4]; used for pushing cycles to the k-skeleton in Theorem 5.3.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Isoperimetric inequalities in Hadamard spaces of asymptotic rank two." pith.science (2026). https://pith.science/paper/CT6KMUS7

@misc{pith2026250604882,
  author       = {Pith},
  title        = {Pith review of: Isoperimetric inequalities in Hadamard spaces of asymptotic rank two},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CT6KMUS7}},
  note         = {Machine review of arXiv:2506.04882}
}
read the original abstract

Gromov's isoperimetric gap conjecture for Hadamard spaces states that cycles in dimensions greater than or equal to the asymptotic rank admit linear isoperimetric filling inequalities, as opposed to the inequalities of Euclidean type in lower dimensions. In the case of asymptotic rank 2, recent progress was made by Dru\c{t}u-Lang-Papasoglu-Stadler who established a homotopical inequality for Lipschitz 2-spheres with exponents arbitrarily close to 1. We prove a homological inequality of the same type for general cycles in dimensions at least 2, assuming that the ambient space has finite linearly controlled asymptotic dimension. This holds in particular for all Hadamard 3-manifolds and finite-dimensional CAT(0) cube complexes.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 47 canonical work pages

  1. [19]

    Goldhirsch, Higher Rank Hyperbolicity and Currents in Metric Spaces, Diss

    T. Goldhirsch, Higher Rank Hyperbolicity and Currents in Metric Spaces, Diss. ETH Zurich No. 29846 (2023)

    work page 2023

  2. [20]

    Goldhirsch, U

    T. Goldhirsch, U. Lang, Characterizations of higher rank hyperbolicity, Math. Z. 305 (2023), Paper No. 13, 26 pp

    work page 2023

  3. [1]

    Adams, W

    S. Adams, W. Ballmann, Amenable isometry groups of Hadamard spaces, Math. Ann. 312 (1998), no. 1, 183–195

    work page 1998

  4. [2]

    F. J. Almgren Jr., Optimal isoperimetric inequalities, Indiana Univ. Math. J. 35 (1986), no. 3, 451–547

    work page 1986

  5. [3]

    Ambrosio, B

    L. Ambrosio, B. Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), 1–80

    work page 2000

  6. [4]

    Arzhantseva, G

    G. Arzhantseva, G. A. Niblo, N. Wright, J. Zhang, A characterization for asymptotic dimension growth, Algebr. Geom. Topol. 18 (2018), no. 1, 493–524

    work page 2018

  7. [5]

    Assouad, Sur la distance de Nagata, C

    P. Assouad, Sur la distance de Nagata, C. R. Acad. Sci. Paris S´ er. I Math. 294 (1982), no. 1, 31-34

    work page 1982

  8. [6]

    Ballmann, Lectures on Spaces of Nonpositive Curvature, DMV Seminar 25, Birkh¨ auser 1995

    W. Ballmann, Lectures on Spaces of Nonpositive Curvature, DMV Seminar 25, Birkh¨ auser 1995

    work page 1995

Show all 48 references
  1. [7]

    Basso, S

    G. Basso, S. Wenger, R. Young, Undistorted fillings in subsets of metric spaces, Adv. Math. 423 (2023), Paper No. 109024, 54 pp

    2023

  2. [8]

    G. Bell, A. Dranishnikov, Asymptotic dimension, Topology Appl. 155 (2008), no. 12, 1265–1296. 20 URS LANG, STEPHAN STADLER, AND DA VID URECH

    2008

  3. [9]

    M. R. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer 1999

    1999

  4. [10]

    Brodskiy, J

    N. Brodskiy, J. Dydak, M. Levin, A. Mitra, A Hurewicz theorem for the Assouad– Nagata dimension, J. Lond. Math. Soc. (2) 77 (2008), no. 3, 741–756

    2008

  5. [11]

    S. V. Buyalo, Capacity dimension and embedding of hyperbolic spaces into the product of trees (Russian), Algebra i Analiz 17 (2005), no. 4, 42–58; translation in St. Petersburg Math. J. 17 (2006), no. 4, 581–591

    2005

  6. [12]

    Buyalo, V

    S. Buyalo, V. Schroeder, Elements of Asymptotic Geometry, EMS Monogr. Math., Europ. Math. Soc. 2007

    2007

  7. [13]

    Dydak, J

    J. Dydak, J. Higes, Asymptotic cones and Assouad–Nagata dimension, Proc. Amer. Math. Soc. 136 (2008), no. 6, 2225–2233

    2008

  8. [14]

    A. N. Dranishnikov, J. Smith, On asymptotic Assouad–Nagata dimension, Topology Appl. 154 (2007), no. 4, 934–952

    2007

  9. [15]

    Drut ¸u, U

    C. Drut ¸u, U. Lang, P. Papasoglu, S. Stadler, Minimal tetrahedra and an isoperimet- ric gap theorem in non-positive curvature, arXiv:2502.03389 [math.MG]

    arXiv

  10. [16]

    Federer, Geometric Measure Theory, Springer 1969

    H. Federer, Geometric Measure Theory, Springer 1969

    1969

  11. [17]

    Fujiwara, P

    K. Fujiwara, P. Papasoglu, Asymptotic dimension of planes and planar graphs, Trans. Amer. Math. Soc. 374 (2021), no. 12, 8887–8901

    2021

  12. [18]

    Goldhirsch, Lipschitz chain approximation of metric integral currents, Anal

    T. Goldhirsch, Lipschitz chain approximation of metric integral currents, Anal. Geom. Metr. Spaces 10 (2022), 40–49

    2022

  13. [21]

    Gromov, Filling Riemannian manifolds, J

    M. Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), 1–147

    1983

  14. [22]

    Gromov, Hyperbolic groups, in: S

    M. Gromov, Hyperbolic groups, in: S. Gersten (ed.), Essays in Group Theory, Math. Sci. Res. Inst. Publ. 8, Springer 1987, pp. 75–263

    1987

  15. [23]

    Gromov, Asymptotic invariants of infinite groups, in: A

    M. Gromov, Asymptotic invariants of infinite groups, in: A. Niblo, M. A. Roller (eds.), Geometric Group Theory, London Math. Soc. Lect. Notes Ser. 182, Cam- bridge Univ. Press 1993, pp. 1–295

    1993

  16. [24]

    Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkh¨ auser 1999

    M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkh¨ auser 1999

    1999

  17. [25]

    Hoffman, J

    D. Hoffman, J. Spruck, Sobolev and isoperimetric inequalities for Riemannian sub- manifolds. Comm. Pure Appl. Math. 27 (1974), 715–727

    1974

  18. [26]

    Isleifsson, Linear isoperimetric inequality for homogeneous Hadamard manifolds, J

    H. Isleifsson, Linear isoperimetric inequality for homogeneous Hadamard manifolds, J. Anal. Topol. (2023), DOI 10.1142/S1793525323500334, 8 pp

    2023 doi

  19. [27]

    Jørgensen, U

    M. Jørgensen, U. Lang, Geodesic spaces of low Nagata dimension, Ann. Fenn. Math. 47 (2022), 83–88

    2022

  20. [28]

    Kleiner, The local structure of length spaces with curvature bounded above, Math

    B. Kleiner, The local structure of length spaces with curvature bounded above, Math. Z. 231 (1999), 409–456

    1999

  21. [29]

    Kleiner, An isoperimetric comparison theorem, Invent

    B. Kleiner, An isoperimetric comparison theorem, Invent. Math. 108 (1992), no. 1, 37–47

    1992

  22. [30]

    Kleiner, U

    B. Kleiner, U. Lang, Higher rank hyperbolicity, Invent. Math. 221 (2020), no. 2, 597–664

    2020

  23. [31]

    Lang, Higher-dimensional linear isoperimetric inequalities in hyperbolic groups, Internat

    U. Lang, Higher-dimensional linear isoperimetric inequalities in hyperbolic groups, Internat. Math. Res. Notices 2000, no. 13, 709–717

    2000

  24. [32]

    Lang, Local currents in metric spaces, J

    U. Lang, Local currents in metric spaces, J. Geom. Anal. 21 (2011), 683–742

    2011

  25. [33]

    U. Lang, C. Plaut, Bilipschitz embeddings of metric spaces into space forms, Geom. Dedicata 87 (2001), no. 1–3, 285–307

    2001

  26. [34]

    U. Lang, T. Schlichenmaier, Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, Internat. Math. Res. Notices 2005, no. 58, 3625–3655

    2005

  27. [35]

    U. Lang, V. Schroeder, Kirszbraun’s theorem and metric spaces of bounded curva- ture, Geom. Funct. Anal. 7 (1997), 535–560. ISOPERIMETRIC INEQUALITIES IN HADAMARD SPACES 21

    1997

  28. [36]

    Lebedeva, Dimensions of products of hyperbolic spaces (Russian), Algebra i Analiz 19 (2007), no

    N. Lebedeva, Dimensions of products of hyperbolic spaces (Russian), Algebra i Analiz 19 (2007), no. 1, 149–176; translation in St. Petersburg Math. J. 19 (2008), no. 1, 107–124

    2007

  29. [37]

    Leuzinger, Optimal higher-dimensional Dehn functions for some CAT(0) lattices, Groups Geom

    E. Leuzinger, Optimal higher-dimensional Dehn functions for some CAT(0) lattices, Groups Geom. Dyn. 8 (2014), 441–466

    2014

  30. [38]

    Meier, S

    D. Meier, S. Wenger, Quasiconformal almost parametrizations of metric surfaces, arXiv:2106.01256 [math.MG], J. Eur. Math. Soc. (2024), DOI 10.4171/JEMS/1470

    2024 arXiv

  31. [39]

    Roe, Lectures on Coarse Geometry, Univ

    J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser. 31, Amer. Math. Soc. 2003

    2003

  32. [40]

    Schulze, Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds, Geom

    F. Schulze, Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds, Geom. Funct. Anal. 30 (2020), no. 1, 255–288

    2020

  33. [41]

    Stadler, The structure of minimal surfaces in CAT(0) spaces, J

    S. Stadler, The structure of minimal surfaces in CAT(0) spaces, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 11, 3521–3554

    2021

  34. [42]

    Wenger, Isoperimetric inequalities of Euclidean type in metric spaces, Geom

    S. Wenger, Isoperimetric inequalities of Euclidean type in metric spaces, Geom. Funct. Anal. 15 (2005), 534–554

    2005

  35. [43]

    Wenger, Filling invariants at infinity and the Euclidean rank of Hadamard spaces, Internat

    S. Wenger, Filling invariants at infinity and the Euclidean rank of Hadamard spaces, Internat. Math. Res. Notices 2006, Art. ID 83090, 33 pp

    2006

  36. [44]

    Wenger, Gromov hyperbolic spaces and the sharp isoperimetric constant, Invent

    S. Wenger, Gromov hyperbolic spaces and the sharp isoperimetric constant, Invent. Math. 171 (2008), 227–255

    2008

  37. [45]

    Wenger, The asymptotic rank of metric spaces, Comment

    S. Wenger, The asymptotic rank of metric spaces, Comment. Math. Helv. 86 (2011), 247–275

    2011

  38. [46]

    Wright, Finite asymptotic dimension for CAT(0) cube complexes, Geom

    N. Wright, Finite asymptotic dimension for CAT(0) cube complexes, Geom. Topol. 16 (2012), no. 1, 527–554

    2012

  39. [47]

    Xie, Nagata dimension and quasi-M¨ obius maps, Conform

    X. Xie, Nagata dimension and quasi-M¨ obius maps, Conform. Geom. Dyn. 12 (2008), 1–9

    2008

  40. [48]

    Z¨ ust, Functions of bounded fractional variation and fractal currents, Geom

    R. Z¨ ust, Functions of bounded fractional variation and fractal currents, Geom. Funct. Anal. 29 (2019), no. 4, 1235–1294. Department of Mathematics, ETH Z¨urich, R¨amistrasse 101, 8092 Z¨urich, Switzerland Email address: lang@math.ethz.ch Max Planck Institute for Mathematics,...

    2019

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.