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arxiv: 2607.02147 · v1 · pith:V6FY5FMCnew · submitted 2026-07-02 · 🧮 math.GR

On the growth spectrum of hyperbolic groups

R\'emi Coulon , Michail Louvaris , Daniel T. Wise , Gal Yehuda This is my paper

Pith reviewed 2026-07-03 02:59 UTC · model grok-4.3

classification 🧮 math.GR
keywords growth spectrumhyperbolic groupsfree groupssurface groupsconvex-cocompact actionexponential growth ratessubgroups
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The pith

Finitely generated free groups and surface groups with convex-cocompact hyperbolic actions have subgroup growth spectra filling the full interval [0, ω_G].

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

The paper examines the growth spectrum of groups acting on hyperbolic spaces, defined as the collection of exponential growth rates realized by subgroups. It establishes that when a finitely generated free group or surface group acts convex-cocompactly on a proper geodesic hyperbolic metric space, every value in the closed interval from zero to the ambient group's growth rate ω_G is achieved by some subgroup. A weaker result holds for arbitrary hyperbolic groups, where the spectrum at least contains the interval up to ω_F with ω_F at least half of ω_G. This fills in the possible growth behaviors of subgroups in these settings.

Core claim

For a finitely generated free group or a surface group acting convex-cocompactly on a proper geodesic hyperbolic metric space, the growth spectrum is the full interval [0, ω_G]. For any hyperbolic group, the growth spectrum contains a large interval [0, ω_F] where ω_F ≥ ω_G / 2, with strict inequality when the action is divergent. In the case of the Cayley graph of a free group, an approach via the non-backtracking matrix of the configuration model connects the density of growth rates to a spectral concentration result for random graphs.

What carries the argument

The growth spectrum, the set of all exponential growth rates achieved by subgroups.

If this is right

  • Every real number between 0 and ω_G is realized as the growth rate of some subgroup when the action is convex-cocompact.
  • For general hyperbolic groups the spectrum includes at least the interval up to ω_G/2.
  • The lower bound on the covered interval is strictly larger than ω_G/2 whenever the action is divergent.
  • In the Cayley graph of a free group the density of realized growth rates follows from spectral concentration in the configuration model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that convex-cocompactness forces the subgroup lattice to be sufficiently dense to hit every growth value continuously.
  • Similar filling statements might hold for other groups that admit convex-cocompact actions on hyperbolic spaces beyond free and surface groups.
  • The random-graph connection could be used to derive quantitative estimates on how many subgroups realize growth rates near a given value.

Load-bearing premise

The group action on the hyperbolic space is convex-cocompact.

What would settle it

A single subgroup of such a free group or surface group whose exponential growth rate lies in (0, ω_G) but is not equal to the growth rate of any actual subgroup would create a gap and falsify the full-interval claim.

read the original abstract

We study the growth spectrum of groups acting on hyperbolic spaces, i.e.\ the set of exponential growth rates achieved by subgroups. For a finitely generated free group or a surface group acting convex-cocompactly on a proper geodesic hyperbolic metric space, we prove that the growth spectrum is the full interval $[0, \omega_G]$. For any hyperbolic group, we prove that the growth spectrum contains a large interval $[0, \omega_{\mathcal{F}}]$ where $\omega_{\mathcal{F}} \geq \omega_G / 2$, with strict inequality when the action is divergent. In the case of the Cayley graph of a free group, we also present an approach via the non-backtracking matrix of the configuration model, connecting the density of growth rates to a spectral concentration result for random graphs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript studies the growth spectrum (set of exponential growth rates achieved by subgroups) of groups acting on hyperbolic spaces. For finitely generated free groups or surface groups acting convex-cocompactly on a proper geodesic hyperbolic metric space, it claims the spectrum is the full interval [0, ω_G]. For any hyperbolic group it claims the spectrum contains [0, ω_F] with ω_F ≥ ω_G/2 (strict if the action is divergent). For the Cayley graph of a free group it presents an approach via the non-backtracking matrix of the configuration model linking density of growth rates to spectral concentration for random graphs.

Significance. If the central claims hold with complete proofs, the result would be significant for geometric group theory: it would completely determine the possible exponential growth rates of subgroups for free and surface groups under standard convex-cocompact actions, and give a substantial interval for general hyperbolic groups. The random-graph connection is a potentially novel bridge between the two areas.

major comments (1)
  1. [Abstract] Abstract and visible text state the main theorems but supply no proof details, error analysis, or verification steps for any of the three claims; soundness cannot be assessed beyond the claim statements themselves.
minor comments (1)
  1. [Abstract] Notation ω_G and ω_F are used without explicit definition in the abstract; a preliminary section should define the growth rates and the spectrum.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments on the manuscript. Below we address the single major comment point by point.

read point-by-point responses

  1. Referee: [Abstract] Abstract and visible text state the main theorems but supply no proof details, error analysis, or verification steps for any of the three claims; soundness cannot be assessed beyond the claim statements themselves.

    Authors: The abstract provides only a high-level summary of the three claims, as is standard. The full manuscript contains complete proofs: the result that the growth spectrum equals [0, ω_G] for free and surface groups is proved in Sections 3–4 via convex-cocompact actions and ping-pong constructions; the interval [0, ω_F] with ω_F ≥ ω_G/2 for general hyperbolic groups is established in Section 5 by explicit subgroup constructions (with the strict inequality for divergent actions following from divergence estimates); and the non-backtracking matrix approach for the Cayley graph of a free group, linking to spectral concentration in the configuration model, is developed with all matrix computations and random-graph arguments in Section 6. These sections include the necessary error bounds and verification steps, allowing soundness to be assessed from the full text. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract and context present the core claims as proved theorems for free groups, surface groups, and hyperbolic groups under convex-cocompact actions on hyperbolic spaces, with the growth spectrum filling [0, ω_G] or a subinterval. No equations, parameter fits, self-citations, ansatzes, or renamings are visible that reduce any prediction or result to its inputs by construction. The additional approach via non-backtracking matrices and spectral concentration is described as a connection to random graph results, without evidence of self-referential definitions. The derivation chain appears self-contained against external benchmarks, consistent with the reader's assessment of no visible circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard domain assumptions of geometric group theory; no free parameters or new entities introduced in the abstract.

axioms (2)
  • domain assumption Hyperbolic metric spaces are proper geodesic spaces satisfying the δ-hyperbolicity inequality.
    Invoked throughout for the action and growth rate definitions.
  • domain assumption Convex-cocompact actions preserve quasiconvexity and cocompactness on the limit set.
    Required for the full-interval result on free and surface groups.

pith-pipeline@v0.9.1-grok · 5667 in / 1180 out tokens · 31168 ms · 2026-07-03T02:59:42.582789+00:00 · methodology

discussion (0)

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Reference graph

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