Pith. sign in

REVIEW 2 major objections 4 minor 47 references

The cheap embedding principle: Dynamical upper bounds for homology growth

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read This paper establishes that for residually finite groups of type FP_{n+1}, the asymptotic Betti numbers and logarithmic torsion sizes along a chain of finite-index normal subgroups are bounded above by the measured embedding dimension and…

desk verdict The new measured embedding invariants and the dynamical upper bounds for homology growth are real and the central argument holds up; the apparent flaw in Corollary 16.6 is not a flaw. read the letter →

arxiv 2508.01347 v2 pith:QVKG67NY submitted 2025-08-02 math.AT math.DSmath.GRmath.OA

Kevin Li , Clara Loeh , Marco Moraschini , Roman Sauer , Matthias Uschold This is my paper · ORCID
classification math.ATmath.DSmath.GRmath.OA MSC 37A2020J0516S3520E2620E18
keywords measuredembeddingdimensionvolumelogarithmictorsionhomologygrowthBettinumbergradientL2-Bettinumbersweakcontainmentboundedorbitequivalenceprofinitecompletion
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 proves a 'cheap embedding principle' for homology growth: if a group's action on its profinite completion can be embedded into a small algebraic complex, then the group's Betti number gradients and logarithmic torsion homology gradients along finite-index subgroups are bounded by the size of that complex. This gives dynamical upper bounds that work uniformly for integer and finite-field coefficients, and it separates torsion growth estimates from Betti number estimates. The authors show the principle is sharp in concrete cases such as free groups and surface groups, where the measured embedding dimension matches known rank gradients. They also develop a quantitative homological algebra over crossed product rings to make the comparison between dynamical systems and finite-index subgroups precise.

What carries the argument

The central object is the crossed product ring L∞(α,Z) ∗ Γ of a standard measure-preserving action, together with marked projective chain complexes and a quantitative homological algebra of 'almost' complexes and maps. The key identity-like tool is the logarithmic norm lognorm, a refined version of dimension times log of operator norm that controls torsion in cokernels and is compatible with approximate equalities and Gromov–Hausdorff distances between complexes. Deformation and strictification theorems show that any complex close to an adapted one can be turned into an honest chain complex without changing its size much, and the density of cylinder sets in the profinite completion allows passage from dynamical complexes to ones associated with deep enough finite-index subgroups.

What would settle it

For a free group F_d with a residual chain, the paper computes medim^Z_1 of the profinite completion to be d−1; if one could construct a 1-dimensional α-embedding with dimension strictly less than d−1, Theorem 1.2 would force the first Betti gradient below d−1, contradicting the known rank gradient. Alternatively, one could test flatness directly by checking whether L∞(bΓ,Z) ⊗_ZΓ C_* is exact for the standard free resolution of a concrete group such as SL_3(Z) in degree 1.

Watch

Extended reading notes

Core claim

The central discovery is that the asymptotic homological size of a residually finite group along a residual chain is controlled by the size of its profinite completion action, measured through two new invariants: the measured embedding dimension and the measured embedding volume. These invariants are defined as infima over marked projective chain complexes over the crossed product ring L∞(α,Z) ∗ Γ that admit an α-embedding from a free resolution of the trivial module. The main theorem, Theorem 1.2, asserts that the upper Betti gradient is at most the measured embedding dimension whenever Z is the integers or a finite field, and the upper logarithmic torsion gradient is at most the measured embedding volume when Z is the integers. The proof works by approximating any embedding by one adapted to cylinder sets, discretising it to complexes over individual finite-index subgroups, and then using homological retracts together with torsion estimates. The same machinery also yields an upper bound for L2-Betti numbers by the measured embedding dimension.

Load-bearing premise

The proof depends on the function space L∞ of the action being flat over the group ring, so that tensoring a free resolution with it still gives a resolution; if that flatness fails, the retraction arguments behind the inequalities collapse.

Editorial extensions

If this is right

  • For residually finite groups of type FP_{n+1}, upper Betti number gradients over Z and over finite fields, as well as logarithmic torsion homology gradients, are all bounded by dynamical invariants of the profinite completion action.
  • Amenable groups have vanishing measured embedding dimension and volume for every standard action, recovering and refining known vanishings of Betti gradients over every field and of logarithmic torsion growth.
  • For free groups of rank d and surface groups of genus g, the measured embedding dimension in degree 1 of the profinite completion equals d−1 and 2g−2 respectively, exactly matching L2-Betti numbers and rank gradients.
  • L2-Betti numbers of a group of type FP_{n+1} are always bounded by the measured embedding dimension of any standard action, giving a dynamical upper bound independent of the action's choice.
  • Measured embedding dimension and volume are multiplicative under weak bounded orbit equivalence, yielding proportionality results for hyperbolic 3-manifolds where these invariants scale with volume.

Reading between the lines

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

  • The decoupling of torsion growth from Betti growth suggests that vanishing of mevol can be verified independently of L2-Betti numbers, which may allow torsion growth vanishing proofs in cases where L2-Betti numbers are non-zero, such as products with amenable factors.
  • Since medim and mevol are monotone under weak containment, the framework points toward a dynamical strategy for the conjecture replacing Q-rank by R-rank in semisimple Lie groups: construct cheap embeddings for the relevant lattice actions rather than computing gradients directly.
  • The cylinder-set approximation underlying the main theorem implies that medim and mevol of a profinite completion can be approximated numerically by looking at deep enough finite-index subgroups, offering a computable route to upper bounds on homology growth.
  • If the measured embedding invariants behave like cost and integral foliated simplicial volume for ergodic decompositions, one would expect medim and mevol of an arbitrary action to equal the essential supremum over its ergodic components, which the paper only establishes in an approximate, one-sided form.
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 proposes two new dynamical invariants, the measured embedding dimension medim_n^Z(α) and the measured embedding volume mevol_n(α), defined as infima over projective chain complexes over crossed product rings admitting an α-embedding. It then proves upper bounds for Betti-number gradients and logarithmic torsion-homology gradients along residual chains of finite-index normal subgroups in terms of these invariants for the profinite completion action (Theorem 1.2). The proof builds an elaborate quantitative homological algebra framework: norms and supports, almost-equality, a Gromov–Hausdorff distance for marked projective complexes, strictification and deformation theorems, and a passage from adapted dynamical embeddings to homological retracts over finite-index subgroups. The paper also states an upper bound on L2-Betti numbers (Theorem 1.3), monotonicity under weak containment, invariance under weak bounded orbit equivalence, and calculations for amenable, free, surface, product, and finite-index-subgroup situations, as well as comparisons with cost and integral foliated simplicial volume.

Significance. If the main estimates hold, the paper creates a genuinely new bridge between measured group theory and homology growth: homology gradients are bounded by an infimum over essentially free dynamical systems, and the invariants have good inheritance properties such as weak-containment monotonicity and weak bounded orbit equivalence invariance. The proofs are unusually explicit, with the full approximation chain from dynamical embeddings to finite-index-retract estimates written out in Sections 3–8. I found the main chain supporting Theorem 1.2 convincing: Proposition 2.10 supplies the needed flatness base change, Theorem 7.6 constructs the retractions, and Theorems 8.4 and 8.5 convert them into gradient estimates. There are no fitted parameters or circular reductions. My concerns are localized to the finite-field scope of the L2-Betti statement and to a missing component in one translation-invariance constant used for weak containment.

major comments (2)
  1. [Theorem 8.6 / Section 8.2] The proof of Theorem 8.6 is only valid when the coefficient ring Z is the ring of integers. For Z = F_p, the expression N R ⊗_{ZΓ} C_* used in the proof is not defined: N R is a complex von Neumann algebra and there is no unital ring homomorphism F_pΓ → N R. Consequently the claim b_n^{(2)}(Γ) ≤ medim^{F_p}_n(α) is not proved. This is not merely cosmetic, because Propositions 12.3(iii) and 12.4(iii) invoke Theorem 8.6 with Z allowed to be a finite field. Please either restrict Theorem 1.3/8.6 to integer coefficients and give a separate argument for the finite-field lower bounds in Section 12 (for free and surface groups such bounds can be obtained from Theorem 1.2 together with EMD* and weak containment, but this is not written), or provide a genuine finite-field proof.
  2. [Theorem 15.29 / Section 15.5] The translation-invariant constant κδ(f_*) in Theorem 15.29 is defined as a maximum involving Q(f0),...,Q(fn), but an n-chain map (Definition 4.3) has components f0,...,f_{n+1}, and κ_n(f_*) from Definition 4.14 includes ∥f_{n+1}∥. As written, the inequality κ_n(f_*) ≤ κδ(f_*) used to apply Theorem 4.15 is not justified. This is a local but real gap in the proof of weak-containment monotonicity (Theorem 15.30). It is repaired by adding Q(f_{n+1}) to the defining maximum and adjusting the constants in Lemma 15.32 and Theorem 15.30 accordingly.
minor comments (4)
  1. [Definition 4.7] The quantity ν_n(D_*) is listed twice with identical definition; one of the two entries appears intended to be a different quantity, perhaps one involving N2 or a distinct norm. Please correct the typo and ensure all subsequent references use the intended quantity.
  2. [Theorem 1.4 and Section 11] The invariant mevol_n is introduced only for integer coefficients, but statements such as Theorem 1.4 and Proposition 1.5(i) assert 'mevol_n(α)=0' without repeating the coefficient convention. Please make explicit in each statement that mevol is an integer-coefficient invariant, so readers do not infer a finite-field version.
  3. [Section 6.2, Proposition 6.4] The function log^+ is used for zero norms in Proposition 6.4(iv), where K can be 0, but log^+ is never defined at 0. Please state the convention log^+(0)=0 or restrict K to positive values.
  4. [Corollary 16.6] Corollary 16.6 is correct as written: for α×β with β trivial, the stabilizer of (x,y) equals the stabilizer of x, so essential freeness of α passes to the diagonal action. A one-sentence justification of this point would preempt reader confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dynamical upper-bound theorem is proven by approximation and retraction from an independently defined infimum; self-citations are motivational only.

full rationale

The central claim Theorem 1.2 is not circular. The measured embedding dimension and measured embedding volume are defined as infima over all alpha-embeddings (Definition 1.1), with no reference to homology gradients. The proof of Theorem 8.1 fixes an arbitrary embedding and derives the gradient bounds through Theorem 8.3 (deformation to Gamma*-adapted embeddings), Theorem 8.4 (homotopy retracts after discretisation), and Theorem 8.5 (lognorm torsion estimates). The retract in Theorem 7.6 uses the fundamental lemma of homological algebra and flatness of the relevant crossed product ring over ZGamma; the only nontrivial input is Proposition 2.10, where flatness of L^infty(alpha)*Gamma over ZGamma follows from the canonical isomorphism (L^infty(alpha)*Gamma) tensor_{ZGamma} M congruent L^infty(alpha) tensor_Z M and the cited freeness of L^infty(alpha) as an abelian group [Ste85]. For finite fields the flatness is immediate. This is an external algebraic fact, not an assumption of the desired inequality. The L2-Betti estimate in Theorem 8.6 is likewise a retraction argument using the same flatness. No fitted parameter is renamed as a prediction: the only infimum is the defining infimum of medim/mevol, and the theorem proves that the gradient is bounded above by every admissible D_n, which is the substantive content of the result. Self-citations to [LLM+] in Remark 1.6 and Section 14.2 are motivational and are used for optional cheap-rebuilding examples, not for the main upper bound. The concern about Corollary 16.6 does not identify circularity: for an essentially free action alpha and a trivial action beta, the stabilizer of (x,y) under the diagonal action is trivial for mu-almost every x, so alpha x beta is a standard action in the paper's sense. No step in the derivation reduces to its own inputs.

Assumptions & free parameters 0 free parameters · 8 assumptions · 2 invented entities

The central inequalities are not the result of fitting: the invariants are defined by infima, and the upper bounds are proven through approximation and retraction. The paper relies on standard theorems for flatness, ergodic decomposition, torsion estimates, Rokhlin lemmas, and weak containment; none are ad hoc inventions, and no free parameters appear. The new invariants are defined entities with external constraints, not fitted to the target bounds.

assumptions (8)
  • standard math L^infty(alpha,Z) is flat over Z; for Z=Z it is free abelian, cited to [Ste85].
    Invoked in Proposition 2.10 to show L^infty(alpha)*Gamma is flat over ZGamma, which underlies the base-change and induction arguments in Sections 7 and 8.
  • standard math Fundamental lemma of homological algebra for projective resolutions.
    Used in Theorem 7.6 and Theorem 8.6 to construct chain maps g_* with g_* composed with f_* homotopic to the identity.
  • standard math Gabber's logarithmic torsion estimate for cokernels, cited to [Sou99, Sau16].
    Gives the upper bound on log of torsion in cokernels used in Theorem 7.7.
  • standard math Ergodic decomposition exists for standard actions, cited to [Var63].
    Used in Section 16 to reduce to ergodic actions and in Corollary 16.4.
  • domain assumption Cylinder sets in the profinite completion form a mu-dense, Gamma-invariant subalgebra.
    This is the bridge in Theorem 5.10 and Theorem 8.3 that allows approximation by finite-index subgroup data.
  • domain assumption Rokhlin lemma for amenable group actions, cited to [CJK+18, Theorem 3.6].
    Used in Theorem 11.11 to construct cheap embeddings for amenable groups.
  • standard math Weak containment is characterized by approximation in weak neighbourhoods up to isomorphism, cited to [Kec10, Proposition 10.1].
    Used in the proof of Theorem 15.30 to transfer embeddings from alpha to beta.
  • domain assumption For infinite group actions there exist cofinite subsets of arbitrarily small measure, cited to [Lev95, Proposition 1].
    Used in Proposition 10.1 and Remark 18.5 to construct degree-zero embeddings and truncations.
invented entities (2)
  • measured embedding dimension medim_n^Z(alpha) independent evidence
    purpose: Dynamical invariant bounding Betti number gradients and L2-Betti numbers.
    Defined as an infimum over alpha-embeddings; Theorem 1.2 proves it bounds independently defined gradients, and Proposition 12.3 computes it as d-1 for free group profinite actions.
  • measured embedding volume mevol_n(alpha) independent evidence
    purpose: Dynamical invariant bounding logarithmic torsion homology growth.
    Theorem 1.2 and Theorem 1.9 give external constraints on its values; it is not a free parameter fitted to data.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The cheap embedding principle: Dynamical upper bounds for homology growth." pith.science (2026). https://pith.science/paper/QVKG67NY

@misc{pith2026250801347,
  author       = {Pith},
  title        = {Pith review of: The cheap embedding principle: Dynamical upper bounds for homology growth},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QVKG67NY}},
  note         = {Machine review of arXiv:2508.01347}
}
read the original abstract

We provide upper bounds for logarithmic torsion homology growth and Betti number growth of groups, phrased in the language of measured group theory.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 45 canonical work pages

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION output.nonempty.mrnumber FUNCTION fin.entry add.period write mrnumber output.nonempty.mrnumber newline INTEGERS nameptr namesleft numnames FUNCTION format.language language empty "" " (" language * ")" * if FUNCTION format.names 's := #1 'nameptr := s num.names 'numnames := numnames 'namesleft := nam...

    work page

  2. [2]

    Ab\'ert, N

    M. Ab\'ert, N. Bergeron, M. Fr a czyk, and D. Gaboriau, On homology torsion growth, J. Eur.\ Math.\ Soc.\ (JEMS) 27 (2025), no. 6, 2293--2357

    work page 2025

  3. [3]

    Agol, Virtual properties of 3-manifolds, Proceedings of the I nternational C ongress of M athematicians--- S eoul 2014

    I. Agol, Virtual properties of 3-manifolds, Proceedings of the I nternational C ongress of M athematicians--- S eoul 2014. V ol. 1, Kyung Moon Sa, Seoul, 2014, pp. 141--170

    work page 2014

  4. [4]

    Ab\'ert and N

    M. Ab\'ert and N. Nikolov, Rank gradient, cost of groups and the rank versus H eegaard genus problem , J. Eur.\ Math.\ Soc.\ (JEMS) 14 (2012), no. 5, 1657--1677

    work page 2012

  5. [5]

    Ab \'e rt and B

    M. Ab \'e rt and B. Weiss, Bernoulli actions are weakly contained in any free action, Ergodic Theory Dyn.\ Syst. 33 (2013), no. 2, 323--333

    work page 2013

  6. [6]

    P. J. Burton and A. S. Kechris, Weak containment of measure-preserving group actions, Ergodic Theory Dynam.\ Systems 40 (2020), no. 10, 2681--2733

    work page 2020

  7. [7]

    M. B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces, London Mathematical Society Lecture Note Series, vol. 269, Cambridge University Press, Cambridge, 2000

    work page 2000

  8. [8]

    Braun and R

    S. Braun and R. Sauer, Volume and macroscopic scalar curvature, Geom.\ Funct.\ Anal. 31 (2021), no. 6, 1321--1376

    work page 2021

Show all 47 references
  1. [9]

    Bowen and R

    L. Bowen and R. D. Tucker-Drob, On a co-induction question of K echris , Israel J.\ Math. 194 (2013), no. 1, 209--224

    2013

  2. [10]

    Bergeron and A

    N. Bergeron and A. Venkatesh, The asymptotic growth of torsion homology for arithmetic groups, J. Inst.\ Math.\ Jussieu 12 (2013), no. 2, 391--447

    2013

  3. [11]

    Cheeger and M

    J. Cheeger and M. Gromov, L_2 -cohomology and group cohomology , Topology 25 (1986), no. 2, 189--215

    1986

  4. [12]

    C. T. Conley, S. C. Jackson, D. Kerr, A. S. Marks, B. Seward, and R. D. Tucker-Drob, F lner tilings for actions of amenable groups, Math.\ Ann. 371 (2018), no. 1-2, 663--683

    2018

  5. [13]

    Fauser, C

    D. Fauser, C. L \"o h, M. Moraschini, and J. P. Quintanilha, Stable integral simplicial volume of 3 -manifolds, J. Topol. 14 (2021), no. 2, 608--640

    2021

  6. [14]

    Frigerio, C

    R. Frigerio, C. L \"o h, C. Pagliantini, and R. Sauer, Integral foliated simplicial volume of aspherical manifolds, Israel J.\ Math. 216 (2016), no. 2, 707--751

    2016

  7. [15]

    Fr a czyk, S

    M. Fr a czyk, S. Mellick, and A. Wilkens, Poisson-voronoi tessellations and fixed price in higher rank, Preprint, arXiv:2307.01194, 2023

    2023 arXiv

  8. [16]

    Furman, Orbit equivalence rigidity, Ann

    A. Furman, Orbit equivalence rigidity, Ann. of Math. (2) 150 (1999), no. 3, 1083--1108

    1999

  9. [17]

    Gaboriau, Co \^u t des relations d' \'e quivalence et des groupes , Invent.\ Math

    D. Gaboriau, Co \^u t des relations d' \'e quivalence et des groupes , Invent.\ Math. 139 (2000), no. 1, 41--98

    2000

  10. [18]

    (2002), no

    , Invariants ^2 de relations d'\'equivalence et de groupes , Publ.\ Math.\ Inst.\ Hautes \'Etudes Sci. (2002), no. 95, 93--150

    2002

  11. [19]

    167--186

    , On orbit equivalence of measure preserving actions, Rigidity in dynamics and geometry ( C ambridge, 2000), Springer, Berlin, 2002, pp. 167--186

    2000

  12. [20]

    A. S. Kechris, Global aspects of ergodic group actions, Mathematical Surveys and Monographs, vol. 160, American Mathematical Society, Providence, RI, 2010

    2010

  13. [21]

    , Weak containment in the space of actions of a free group, Israel J. Math. 189 (2012), 461--507

    2012

  14. [22]

    A. Kar, P. Kropholler, and N. Nikolov, On growth of homology torsion in amenable groups, Math. Proc. Cambridge Philos. Soc. 162 (2017), no. 2, 337--351

    2017

  15. [23]

    A. S. Kechris and B. D. Miller, Topics in orbit equivalence, Lecture Notes in Mathematics, vol. 1852, Springer, 2004

    2004

  16. [24]

    T. T. Q. L \^e , Growth of homology torsion in finite coverings and hyperbolic volume, Ann.\ Inst.\ Fourier (Grenoble) 68 (2018), no. 2, 611--645

    2018

  17. [25]

    Levitt, On the cost of generating an equivalence relation, Ergodic Theory Dynam.\ Systems 15 (1995), no

    G. Levitt, On the cost of generating an equivalence relation, Ergodic Theory Dynam.\ Systems 15 (1995), no. 6, 1173--1181

    1995

  18. [26]

    K. Li, C. L\"oh, M. Moraschini, R. Sauer, and M. Uschold, The algebraic cheap rebuilding property, Preprint, arXiv:2409.05774, 2024

    2024

  19. [27]

    Linnell, W

    P. Linnell, W. L\"uck, and R. Sauer, The limit of F_p - B etti numbers of a tower of finite covers with amenable fundamental groups , Proc.\ Amer.\ Math.\ Soc. 139 (2011), no. 2, 421--434

    2011

  20. [28]

    L \"u ck and T

    W. L \"u ck and T. Macko, Surgery theory. F oundations , Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 362, Springer, Cham, 2024

    2024

  21. [29]

    L \"o h, M

    C. L \"o h, M. Moraschini, and R. Sauer, Amenable covers and integral foliated simplicial volume, New York J.\ Math. 28 (2022), 1112--1136

    2022

  22. [30]

    L \"o h, Cost vs.\ integral foliated simplicial volume, Groups Geom.\ Dyn

    C. L \"o h, Cost vs.\ integral foliated simplicial volume, Groups Geom.\ Dyn. 14 (2020), no. 3, 899--916

    2020

  23. [31]

    , Ergodic theoretic methods in group homology---a minicourse on L^2 - B etti numbers in group theory , SpringerBriefs in Mathematics, Springer, Cham, 2020

    2020

  24. [32]

    L \"o h and C

    C. L \"o h and C. Pagliantini, Integral foliated simplicial volume of hyperbolic 3-manifolds, Groups Geom.\ Dyn. 10 (2016), no. 3, 825--865

    2016

  25. [33]

    L\"oh and G

    C. L\"oh and G. Sartori, Integral foliated simplicial volume and ergodic decomposition, Ann.\ Math.\ Blaise Pascal 31 (2024), no. 1, 47--64

    2024

  26. [34]

    L\"uck , R

    W. L\"uck , R. Sauer, and C. Wegner, L^2 -torsion, the measure-theoretic determinant conjecture, and uniform measure equivalence , J. Topol.\ Anal. 2 (2010), no. 2, 145--171

    2010

  27. [35]

    L \"u ck, Approximating L^2 -invariants by their finite-dimensional analogues , Geom.\ Funct.\ Anal

    W. L \"u ck, Approximating L^2 -invariants by their finite-dimensional analogues , Geom.\ Funct.\ Anal. 4 (1994), no. 4, 455--481

    1994

  28. [36]

    , Approximating L^2 -invariants and homology growth , Geom. Funct. Anal. 23 (2013), no. 2, 622--663

    2013

  29. [37]

    D. S. Ornstein and B. Weiss, Ergodic theory of amenable group actions. I . T he R ohlin lemma , Bull.\ Amer.\ Math.\ Soc.\ (N.S.) 2 (1980), no. 1, 161--164

    1980

  30. [38]

    a lische Wilhelms-Universit\

    R. Sauer, L ^ 2 -invariants of groups and discrete measured groupoids , Ph.D. thesis, Westf \"a lische Wilhelms-Universit\"at M \"u nster, 2003, https://nbn-resolving.de/urn:nbn:de:hbz:6-85659549583

    2003

  31. [39]

    Algebra Comput

    , L^2 - B etti numbers of discrete measured groupoids , Internat.\ J. Algebra Comput. 15 (2005), no. 5-6, 1169--1188

    2005

  32. [40]

    16 (2006), no

    , Homological invariants and quasi-isometry, Geom.\ Funct.\ Anal. 16 (2006), no. 2, 476--515

    2006

  33. [41]

    Reine Angew.\ Math

    , Amenable covers, volume and L^2 - B etti numbers of aspherical manifolds , J. Reine Angew.\ Math. 636 (2009), 47--92

    2009

  34. [42]

    20 (2016), no

    , Volume and homology growth of aspherical manifolds, Geom.\ Topol. 20 (2016), no. 2, 1035--1059

    2016

  35. [43]

    a lische Wilhelms-Universit\

    M. Schmidt, L ^2 - B etti N umbers of R - S paces and the I ntegral F oliated S implicial V olume , Ph.D. thesis, Westf \"a lische Wilhelms-Universit\"at M \"u nster, 2005, http://nbn-resolving.de/urn:nbn:de:hbz:6-05699458563

    2005

  36. [44]

    Soul \'e , Perfect forms and the V andiver conjecture , J

    C. Soul \'e , Perfect forms and the V andiver conjecture , J. Reine Angew.\ Math. 517 (1999), 209--221

    1999

  37. [45]

    Stepr \=a ns, A characterization of free abelian groups, Proc.\ Amer.\ Math.\ Soc

    J. Stepr \=a ns, A characterization of free abelian groups, Proc.\ Amer.\ Math.\ Soc. 93 (1985), no. 2, 347--349

    1985

  38. [46]

    R. D. Tucker-Drob, Weak equivalence and non-classifiability of measure preserving actions, Ergodic Theory Dynam.\ Systems 35 (2015), no. 1, 293--336

    2015

  39. [47]

    V. S. Varadarajan, Groups of automorphisms of B orel spaces , Trans.\ Amer.\ Math.\ Soc. 109 (1963), 191--220

    1963

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.