Torus covers with controlled volume and diameter

Sergio Zamora

REVIEW 1 major objections 4 minor 28 references

Torus covers with controlled volume and diameter

T0 review · 1 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 For every Ricci-bounded torus of bounded diameter, a finite cover with controlled volume and diameter exists.

desk verdict A genuinely useful repair of a known gap, with a new covering construction; the main proof is close, but one condition for Theorem 20 is asserted rather than verified. read the letter →

arxiv 2506.00763 v2 pith:ON52PDDF submitted 2025-06-01 math.DG math.MG

Sergio Zamora This is my paper

classification math.DGmath.MG MSC 53C2353C2157M10

keywords RiemanniantoriRiccicurvaturelowerboundfinitecoveringspacesnoncollapsedcoversGromov–HausdorffconvergenceZ^nlatticeactionsvolumeanddiameterbounds

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

The paper establishes Theorem 1: for each dimension $n$ and diameter bound $D$, every Riemannian manifold homeomorphic to the $n$-torus with $\mathrm{Ric}(M)\ge -(n-1)$ and $\mathrm{diam}(M)\le D$ has a finite-sheeted cover whose volume is bounded below by a constant $\varepsilon(n)>0$ and whose diameter is bounded above by $D'(n,D)$. The force of the result is uniformity: neither the volume lower bound nor the diameter upper bound is allowed to degrade as the metric varies, so every such torus is, in a quantitative sense, finitely covered by a noncollapsed space. The proof works on the universal cover, where the fundamental group acts as $\mathbb{Z}^n$; a general theorem about such actions on precompact classes of pointed geodesic spaces produces a subgroup with a displacement gap and a controlled quotient, and a volume estimate for universal covers of aspherical manifolds turns that displacement gap into a volume lower bound. This repairs a gap in an earlier torus-cover theorem and restores a stability theorem for tori under Gromov–Hausdorff convergence.

What carries the argument

The engine of the paper is Theorem 20, a monodromy-like covering-space construction. Given a proper geodesic space $X$, a closed group $\Gamma$ of isometries, a connected open set $B\subset X$ with $\Gamma B=X$, and a symmetric generating set $S\subset\Gamma$ with $T=S^M$, the theorem assembles an abstract group $\tilde\Gamma$ generated by $T^3$ with the relations inherited from $\Gamma$, and then builds a space $\tilde X$ by gluing copies of $B$ along the action of $T$. Under three overlap conditions on $B$ and $T$, the natural map $\Psi:\tilde X\to X$ is a covering map, and it is a homeomorphism exactly when every element of $\tilde\Gamma$ that moves $B$ into itself already lies in $T^\sharp$. In the proof of Theorem 5 the construction is applied to a non-faithful $\mathbb{Z}^N$-action on simply connected spaces $X_i$ that emerges from an equivariant Gromov–Hausdorff limit; the non-faithfulness makes $\Psi$ nontrivial, contradicting simple connectivity. The earlier rank-increase step is powered by Lemma 17, a structure theorem for locally compact abelian groups, which produces a co-compact discrete subgroup $H\cong\mathbb{Z}^m$ in the limit group and raises the rank from $n$ to $N>n$.

What would settle it

Compute, for a sequence of $n$-dimensional tori with $\mathrm{Ric}\ge -(n-1)$ and $\mathrm{diam}\le D$, the maximum volume of a unit ball in the universal cover; if this maximum tends to $0$ along the sequence, then the volume bound of Theorem 1 cannot hold and the theorem would be false.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1, is that for every $n\in\mathbb{N}$ and $D>0$ there exist $\varepsilon(n)>0$ and $D'(n,D)>0$ such that any Riemannian manifold $M$ homeomorphic to the $n$-dimensional torus with $\mathrm{Ric}(M)\ge -(n-1)$ and $\mathrm{diam}(M)\le D$ admits a covering space $M'\to M$ with $\mathrm{vol}(M')\ge \varepsilon$ and $\mathrm{diam}(M')\le D'$ under the lifted metric. The proof reduces the problem to a statement about lattice actions on universal covers. The universal cover $\tilde M$ is a complete simply connected Riemannian manifold with $\mathrm{Ric}\ge -(n-1)$, and $\pi_1(M)\cong\mathbb{Z}^n$ acts on it by deck transformations. A general theorem (Theorem 3) applies to this action because the class of such pointed spaces is precompact in the pointed Gromov–Hausdorff topology: it yields a finite-index subgroup $\Gamma\le\pi_1(M)$ with $\mathrm{diam}(\tilde M/\Gamma)\le D'$ and $d(gp,p)\ge D$ for every non-identity $g\in\Gamma$. The displacement gap ensures the ball $B_{D/2}(p)$ injects into the quotient $M'=\tilde M/\Gamma$, and the volume of that ball is controlled from below by a volume estimate for universal covers of closed aspherical manifolds. The result partially recovers a previously claimed theorem whose proof contained a false statement about Poincaré duality cycles; the new proof avoids that construction entirely.

Load-bearing premise

The proof needs the deep volume estimate of Theorem 40: the universal cover of every closed aspherical Riemannian manifold contains a unit ball whose volume is at least a constant depending only on the dimension.

Editorial extensions

If this is right

The stability theorem for tori under Gromov–Hausdorff convergence with a lower sectional curvature bound now has a complete proof, because Theorem 1 supplies exactly the ingredient that was missing.
In dimension 4, the corresponding torus-stability result under a two-sided Ricci bound also goes through, since it had relied on the same flawed input.
No analogue of Theorem 1 can hold for nilmanifolds or other closed manifolds with torsion-free, non-virtually-abelian fundamental groups; Proposition 8 shows the existence of such covers forces the fundamental group to be virtually abelian.
If Conjecture 9 is true, the Ricci curvature hypothesis in Theorem 1 can be dropped entirely, because the conclusion would follow from pure metric-geometric data about simply connected geodesic spaces with co-compact $\mathbb{Z}^n$-actions.
When the simple-connectivity hypothesis in Theorem 3 is replaced by a bound on the asymptotic volume of the $\mathbb{Z}^n$-action, the same conclusion follows (Theorem 12), giving a stable-norm route to controlled covers.

Reading between the lines

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

The monodromy construction of Theorem 20 looks reusable beyond the paper's setting: any equivariant limit that produces a non-faithful co-compact higher-rank action on a simply connected space should force a covering-space contradiction, so the method may transfer to collapsing or nilpotent-limit problems where the limiting group has larger rank.
A natural quantitative question the paper does not address is the optimal dependence of $\varepsilon(n)$ and $D'(n,D)$ in Theorem 1; the proof is non-effective, passing through compactness and contradiction, so explicit bounds would be a testable next step.
Example 35 shows simple connectivity is essential in Theorem 5; identifying the minimal topological condition on $X_i$ under which the diameter blow-up is impossible would sharpen the boundary between the torus theorem and the nilmanifold counterexamples.
Theorem 3 is stated for precompact classes of geodesic spaces; since the Ricci lower bound is used only for precompactness, any curvature condition or structural hypothesis that implies precompactness would yield the same cover theorem.

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.

Desk Editor's Note

read the letter

Read the whole thing. The paper does two real things: it explains clearly why the Kloeckner-Sabourau proof fails, and it proves a flexible covering-space construction (Theorem 20) plus a precompactness argument (Theorem 5) that recovers the Brué-Naber-Semola torus stability. The abstract group construction in Section 4 is new and useful; the fact that you can tune B, S, T rather than have them forced by the action is what makes the later argument go. The exposition is honest about the reliance on Guth's volume bound (Theorem 40), which is deep but correctly applied.

The soft spot is in the proof of Theorem 5. When Theorem 20 is invoked on the tuple (X_i, H_i, B_i, S, T), one of its hypotheses is ΓB = X, meaning here H_i B_r(p_i) = X_i. Lemma 37 checks conditions (i)-(iv), but not this covering-radius condition. This is not a freebie: for a subgroup H_i of a co-compact group G_i, diam(X_i/H_i) can be much larger than diam(X_i/G_i) (Example 35 is exactly that phenomenon). The condition does follow from the equivariant convergence already established: (X_i, H_i) → (X, H), so diam(X_i/H_i) → diam(X/H), and since r was chosen larger than diam(X/H), for large i we get H_i B_r(p_i) = X_i. But the written proof never says this. A referee should ask for the missing sentence. It is a genuine gap as written, but a small one; the argument goes through with a short addition.

Everything else checks out. The proof that Ψ_i is not a homeomorphism from the infinite kernel is sound because Lemma 37(iv) prevents nontrivial kernel elements in T^6, so (22) fails. Theorem 3 follows from Theorem 5 and Gromov's abelianization lemma cleanly. The citation pattern is fine; the self-citation to [25] is not load-bearing.

Bottom line: this paper deserves a serious referee, and with a small patch to Theorem 5 it should be accepted. I'd bring it to our reading group not because it's flashy but because the covering construction is a good example of how to build covers with controlled diameter from an approximate limit action.

Referee Report

1 major / 4 minor

Summary. The paper proves that for each n ∈ N and D > 0 there exist ε(n) > 0 and D'(n,D) > 0 such that if M is a Riemannian manifold homeomorphic to the n-torus with Ric(M) ≥ −(n−1) and diam(M) ≤ D, then M admits a finite-sheeted covering space M' → M, equipped with the lifted metric, satisfying vol(M') ≥ ε and diam(M') ≤ D'. The proof combines a new general theorem (Theorem 3) about discrete Z^n-actions on pointed proper geodesic spaces, which guarantees a finite-index subgroup with bounded quotient diameter and a lower displacement bound, with Guth's volume estimate for balls in universal covers of aspherical manifolds. The main technical novelty is a covering-space construction (Theorem 20) that builds covers from group actions with flexible choices of the open set B and generating sets S,T. The paper also explains how the main theorem fixes a gap in the proof of the Brué–Naber–Semola stability theorem for tori under sectional curvature bounds.

Significance. If the proof is completed, this is a significant contribution: it provides a new uniform non-collapsing statement for torus covers under only a Ricci lower bound and a diameter bound, and it fills a gap in the literature that had blocked the Brué–Naber–Semola torus stability theorem. The paper is largely self-contained, with explicit lemmas and a careful presentation of the covering construction, and it is honest about its reliance on Guth's deep volume bound. The new Theorem 20 is likely to be of independent interest for constructing covers of metric spaces from group actions. The main concern is a missing verification in the proof of Theorem 5, which is repairable by a standard equivariant Gromov–Hausdorff argument.

major comments (1)

[§5, proof of Theorem 5 (after Lemma 37)] In the paragraph after Lemma 37, the proof asserts that 'by Lemma 37, Theorem 20 applies to the tuple (X_i, H_i, B_i, S, T)' and uses the resulting covering map Ψ_i to contradict the simple-connectedness of X_i. However, Theorem 20 requires, as an explicit hypothesis, that the connected open set B satisfy ΓB = X; here this means H_i B_r(p_i) = X_i. Lemma 37 verifies conditions (i)–(iv) of Theorem 20, but not this covering condition. Conditions (i)–(iii) do not imply it: for example, a disk of radius 0.6 in R^2 with the standard Z^2 translation action satisfies the analogues of (i)–(iii) for suitable S and T, yet the union of its translates does not cover R^2. Consequently, without this verification the map Ψ_i produced by Theorem 20 need not be a covering map onto all of X_i, and the contradiction with the simple-connectedness of X_i is not established. This gap can be repaired: since Φ_i(g) → Φ(g) for all g ∈ Z^N, the triples (X_i, p_i, H_i) converge in the equivariant Gromov–Hausdorff sense to (X, p, H) (after passing to a subsequence), so by Theorem 19 the quotients X_i/H_i converge to X/H; as r > diam(X/H), one obtains diam(X_i/H_i) < r for all large i, hence H_i B_r(p_i) = X_i. The proof should state this argument explicitly.

minor comments (4)

[§1, Abstract] The word 'diam eter' in the abstract is a typo; it should be 'diameter'.
[§3.1, Theorem 15] The exponent in 'S⌊ ℓ+2 3 ⌋' is typeset unclearly; it should presumably be S^{⌊(ℓ+2)/3⌋}.
[§5, proof of Theorem 5] The deduction that G/Γ is not compact from the equality (X/Γ)/(G/Γ) = X/G would benefit from a one-line justification: if G/Γ were compact, a compact fundamental domain C ⊂ G for Γ together with a compact K ⊂ X with GK = X would imply X = CΓK, so X/Γ is the image of the compact set CK and hence compact.
[§6, proof of Theorem 1] It would be clearer to state explicitly that, since D ≥ 2, the ball B_{D/2}(p) contains B_1(p), so the Guth lower bound on vol(B_1(p)) directly gives vol(M') ≥ ε(n).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1 is derived from independent external theorems and a self-contained covering-space construction; self-citations are illustrative only.

full rationale

The derivation chain for Theorem 1 is linear and non-self-referential: Theorem 1 is deduced from Theorem 3 plus Guth's Theorem 40; Theorem 3 is proved from Gromov's Theorem 4 and Theorem 5; and Theorem 5 is proved using equivariant convergence (Fukaya–Yamaguchi), Lemma 17, and the covering-space construction Theorem 20, which is proved in full in Section 4 (Lemmas 27–34). None of these steps defines the target quantity into its own input. The lower volume bound in Theorem 1 comes from Guth's external volume estimate on contractible universal covers of aspherical manifolds, not from assuming the existence of the sought cover. The diameter control comes from Theorem 3, whose proof is independent of Theorem 1 and proceeds by contradiction with a compactness argument. The self-citations to the author's previous work appear only as comparisons or background ('similar to the ones from [9, 21, 25]', 'cf. [25, Section 2.5]', and Example 25's parenthetical), and no load-bearing assertion is imported from the author's prior work; even the cited construction is re-proved as Theorem 20. Proposition 8's connection to [26] is a side remark about non-generalization and is not an input to Theorem 1. The potential concern that the application of Theorem 20 in the proof of Theorem 5 may not explicitly verify the hypothesis ΓB = X is a correctness gap in the written proof, not a circularity: it does not make the conclusion identical to the hypotheses and does not rename a fitted parameter as a prediction. Therefore no circular step meets the required quoted-evidence standard, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof depends on standard tools from metric geometry and geometric group theory. No numbers are fitted to data; the constants eps(n) and D'(n,D) are outputs of the proof, not inputs. Guth's volume estimate is the deepest external input.

assumptions (6)

standard math Gromov precompactness for the class of pointed complete n-manifolds with Ric >= -(n-1).
Used in Section 6 to place universal covers of tori in a precompact class so that Theorem 3 applies.
standard math Theorem 18 (Fukaya-Yamaguchi): after passing to a subsequence, equivariant pointed Gromov-Hausdorff convergence of spaces with closed isometry groups holds.
Used in the proof of Theorem 5 to extract limiting groups Gamma and G from the sequences Gamma_i and G_i.
standard math Lemma 17 (Pontrjagin): a locally compact compactly generated Hausdorff abelian group has a discrete co-compact subgroup isomorphic to Z^m.
Used in the proof of Theorem 5 to find a co-compact lattice H in the quotient G/Gamma_0, giving an unfaithful Z^N action on X_i.
standard math Theorem 15: finite defining sets exist for finitely presented groups; if a symmetric generating set S presents G with relators of length at most l, then S^{floor((l+2)/3)} is a defining set.
Used in Lemma 37 to ensure T^3 is a defining set of Z^N so that the abstract group in Theorem 20 is exactly Z^N.
standard math Theorem 40 (Guth): the universal cover of a closed aspherical n-manifold has a ball of radius r with volume at least eps(n) r^n.
Used at the end of Theorem 1 to produce a basepoint p with vol(B_1(p)) >= eps(n), which is then mapped injectively into the constructed cover.
standard math Theorem 4 (Gromov): in a proper geodesic space with discrete group action and quotient of diameter <= D, there are group elements generating a basis of G_ab tensor R with displacement between D and 2D.
Used in the proof of Theorem 3 to find the finite-index subgroup Gamma_i and its generators.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Torus covers with controlled volume and diameter." pith.science (2026). https://pith.science/paper/ON52PDDF

@misc{pith2026250600763,
  author       = {Pith},
  title        = {Pith review of: Torus covers with controlled volume and diameter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ON52PDDF}},
  note         = {Machine review of arXiv:2506.00763}
}

read the original abstract

We show that under a lower Ricci curvature bound and an upper diameter bound, a torus admits a finite-sheeted covering space with volume bounded from below and diameter bounded from above. This partially recovers a result of Kloeckner and Sabourau, whose original proof contains a serious gap that currently lacks a resolution.

Figures

Figures reproduced from arXiv: 2506.00763 by the authors.

**Figure 1.** X˜ is obtained from copies of B glued one after the other. Lemma 28. X˜ is connected. Proof. Since S j ⊂ T 3 for all j ∈ {1, . . . , 3M}, we have (S j−1 ) ♯S ♯ = (S j ) ♯ for such j’s. Using this, one gets (S ♯ ) 3M = (T 3 ) ♯ , meaning S ♯ generates (T 3 ) ♯ , and consequently Γ. The result ˜ then follows from (i) and the fact that B is connected. Fix p ∈ X and define Σ := {g ∈ Γ˜ | p ∈ Φ(g)B}. Let K ⊂ Σ be a maxim… view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

28 extracted references · 27 canonical work pages

[25]

Limits of almost homogeneous spaces and their fundamental grou ps

Sergio Zamora . Limits of almost homogeneous spaces and their fundamental grou ps. Groups Geom. Dyn. 18 (2024), no. 3, 761798

work page 2024
[1]

Stability of tori under lower sectional curvature

Elia Bru ´ e, Aaron Naber, and Daniele Semola . Stability of tori under lower sectional curvature. Geom. Topol. 28 (2024), no. 8, 39613972

work page 2024
[2]

Periodic metrics.Seminar on Dynamical Systems (St

Dmitri Burago . Periodic metrics.Seminar on Dynamical Systems (St. Petersburg, 1991), 9095. Progr. Nonlinear Diﬀerential Equations Appl., 12 Birkh¨ auser Verlag , Basel, 1994. ISBN:3-7643- 2980-7

work page 1991
[3]

A course in metric geometry

Dmitri Burago, Yuri Burago, and Sergei Ivanov . A course in metric geometry. Grad. Stud. Math., 33. American Mathematical Society, Providence, RI, 2001, xiv+415 pp. ISBN: 0- 8218-2129-6

work page 2001
[4]

A quantitative bounded distance theorem and a Margulis’ lemma for Zn-actions, with applications to homology

Filippo Cerocchi and Andrea Sambusetti . A quantitative bounded distance theorem and a Margulis’ lemma for Zn-actions, with applications to homology. Groups Geom. Dyn. 10 (201 6), no. 4, 12271247

work page
[5]

Tobias H. Colding . Ricci curvature and volume convergence. Ann. of Math. (2) 145 (1997), no. 3, 477501

work page 1997
[6]

Metric geometry of locally compact groups

Yves Cornulier and Pierre de la Harpe . Metric geometry of locally compact groups. Winner of the 2016 EMS Monograph Award EMS Tracts Math., 25 Euro pean Mathematical Society (EMS), Zrich, 2016. viii+235 pp. ISBN:978-3-03719-166-8

work page 2016
[7]

Topics in geometric group theory

Pierre de la Harpe . Topics in geometric group theory. Chicago Lectures in Math. Unive rsity of Chicago Press, Chicago, IL, 2000. vi+310 pp. ISBN:0-226-3171 9-6, ISBN:0-226-31721-8

work page 2000

Show all 28 references

[8]

Fukaya: Theory of convergence for Riemannian orbifolds

K. Fukaya: Theory of convergence for Riemannian orbifolds. Japanese jour nal of mathematics. New series, 12(1) (1986), 121-160

1986
[9]

Fukaya and T

K. Fukaya and T. Yamaguchi : The fundamental groups of almost non-negatively curved manifolds. Ann. of Math. (2) 136 (1992), no.2, 253-333

1992
[10]

Metric structures for Riemannian and non-Riemannian spaces

Misha Gromov . Metric structures for Riemannian and non-Riemannian spaces. Ba sed on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Se mmes. Translated from the French by Sean Michael Bates. Reprint of the 2001 English editio n. Mod. Birkhuser Class. Bir...

1981
[11]

Groups of polynomial growth and expanding maps

Mikhael Gromov . Groups of polynomial growth and expanding maps. Inst. Hautes ´Etudes Sci. Publ. Math. No. 53 (1981), 53-73

1981
[12]

Volumes of balls in large Riemannian manifolds

Larry Guth. Volumes of balls in large Riemannian manifolds. Ann. of Math. (2) 173 ( 2011), no. 1, 5176. 14 SERGIO ZAMORA

2011
[13]

Collapsed mani- folds with Ricci bounded covering geometry

Hongzhi Huang, Lingling Kong, Rong, Xiaochun, and Shicheng Xu. Collapsed mani- folds with Ricci bounded covering geometry. Trans. Amer. Math. S oc. 373 (2020), no. 11, 8039- 8057

2020
[14]

Fibrations and stability for compact group actions on manifolds with local bounded Ricci covering geometry

Hongzhi Huang . Fibrations and stability for compact group actions on manifolds with local bounded Ricci covering geometry. Front. Math. China 15 (2020), no. 1, 69-89

2020
[15]

Nilpotency, almost non- negative curvature, and the gradient ﬂow on Alexandrov spaces

Vitali Kapovitch, Anton Petrunin, and Wilderich Tuschmann . Nilpotency, almost non- negative curvature, and the gradient ﬂow on Alexandrov spaces. Ann. of Math. (2) 171 (2010), no. 1, 343-373

2010
[16]

Vitali Kapovitch and Burkhard Wilking . (2011). Structure of fundamental groups of man- ifolds with Ricci curvature bounded below. arXiv preprint arXiv:1105 .5955

2011
[17]

Kloeckner and S

B. Kloeckner and S. Sabourau . (2020). Mixed sectional-Ricci curvature obstructions on tori. Journal of Topology and Analysis, 12(03), 713-734

2020
[18]

Margulis lemma for compact Lie groups

Marcin Mazur, Xiaochun Rong, and Yusheng W ang . Margulis lemma for compact Lie groups. Math. Z. 258 (2008), no. 2, 395-406

2008
[19]

Milman and Gideon Schechtman

Vitali D. Milman and Gideon Schechtman . Asymptotic theory of ﬁnite-dimensional normed spaces. With an appendix by M. Gromov. Lecture Notes in Math., 120 0. Springer-Verlag, Berlin,
[20]

Topology and ε-regularity theorems on collapsed manifolds with Ricci curvature bounds

Aaron Naber and Ruobing Zhang . Topology and ε-regularity theorems on collapsed manifolds with Ricci curvature bounds. Geom. Topol. 20 (2016), no.5, 2575- 2664

2016
[21]

Some topological results of Ricci limit spaces

Jiayin Pan and Jikang W ang . Some topological results of Ricci limit spaces. Trans. Amer. Math. Soc. 375 (2022), no. 12, 84458464

2022
[22]

Pontrjagin

L. Pontrjagin . Topological groups. Translated from the Russian by Emma Lehmer . (Fifth printing, 1958). Princeton University Press, Princeton, NJ, 1939 . ix+299 pp

1958
[23]

Xiaochun Rong . (2025). Gromov-Hausdorﬀ Limits of Aspherical Manifolds. arXiv p reprint arXiv:2503.23107

2025 arXiv
[24]

Fourier analysis on groups

W alter Rudin. Fourier analysis on groups. Reprint of the 1962 original. Wiley Classic s Lib. Wiley-Intersci. Publ. John Wiley & Sons, Inc., New York, 1990. x+285 pp. ISBN:0-471-52364-X

1962
[26]

Sergio Zamora and Xingyu Zhu . (2024). Topological rigidity of small RCD (K, N) spaces with maximal rank. arXiv preprint arXiv:2406.10189

2024
[27]

Shengxuan Zhou . (2025). On the Gromov-Hausdorﬀ limits of Tori with Ricci condition s. arXiv preprint arXiv:2309.10997v3. Oregon State University

2025 arXiv
[1986]

ISBN:3-540-16769-2

viii+156 pp. ISBN:3-540-16769-2

Pith tools

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

[1] [25]

Limits of almost homogeneous spaces and their fundamental grou ps

Sergio Zamora . Limits of almost homogeneous spaces and their fundamental grou ps. Groups Geom. Dyn. 18 (2024), no. 3, 761798

work page 2024

[2] [1]

Stability of tori under lower sectional curvature

Elia Bru ´ e, Aaron Naber, and Daniele Semola . Stability of tori under lower sectional curvature. Geom. Topol. 28 (2024), no. 8, 39613972

work page 2024

[3] [2]

Periodic metrics.Seminar on Dynamical Systems (St

Dmitri Burago . Periodic metrics.Seminar on Dynamical Systems (St. Petersburg, 1991), 9095. Progr. Nonlinear Diﬀerential Equations Appl., 12 Birkh¨ auser Verlag , Basel, 1994. ISBN:3-7643- 2980-7

work page 1991

[4] [3]

A course in metric geometry

Dmitri Burago, Yuri Burago, and Sergei Ivanov . A course in metric geometry. Grad. Stud. Math., 33. American Mathematical Society, Providence, RI, 2001, xiv+415 pp. ISBN: 0- 8218-2129-6

work page 2001

[5] [4]

A quantitative bounded distance theorem and a Margulis’ lemma for Zn-actions, with applications to homology

Filippo Cerocchi and Andrea Sambusetti . A quantitative bounded distance theorem and a Margulis’ lemma for Zn-actions, with applications to homology. Groups Geom. Dyn. 10 (201 6), no. 4, 12271247

work page

[6] [5]

Tobias H. Colding . Ricci curvature and volume convergence. Ann. of Math. (2) 145 (1997), no. 3, 477501

work page 1997

[7] [6]

Metric geometry of locally compact groups

Yves Cornulier and Pierre de la Harpe . Metric geometry of locally compact groups. Winner of the 2016 EMS Monograph Award EMS Tracts Math., 25 Euro pean Mathematical Society (EMS), Zrich, 2016. viii+235 pp. ISBN:978-3-03719-166-8

work page 2016

[8] [7]

Topics in geometric group theory

Pierre de la Harpe . Topics in geometric group theory. Chicago Lectures in Math. Unive rsity of Chicago Press, Chicago, IL, 2000. vi+310 pp. ISBN:0-226-3171 9-6, ISBN:0-226-31721-8

work page 2000