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arxiv: 2206.06749 · v2 · submitted 2022-06-14 · 🧮 math.GR · math.GT· math.MG

Growth of quasi-convex subgroups in groups with a constricting element

Xabier Legaspi This is my paper

Pith reviewed 2026-05-24 12:18 UTC · model grok-4.3

classification 🧮 math.GR math.GTmath.MG
keywords quasi-convex subgroupsexponential growth ratesconstricting elementpath systemrelatively hyperbolic groupsCAT(0) groupshierarchically hyperbolic groupsMorse element
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The pith

If a group contains a constricting element with respect to a path system, its infinite-index quasi-convex subgroups have relative exponential growth rates strictly smaller than the group's rate.

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

The paper examines the spectrum of relative and quotient exponential growth rates for infinite-index subgroups that are quasi-convex with respect to a chosen path system in a group G acting on a geodesic metric space. When G contains a constricting element relative to the same path system, the relative growth rates of these subgroups fall strictly below the growth rate of G while the quotient growth rates coincide with it. The result applies directly to relatively hyperbolic groups, CAT(0) groups, and hierarchically hyperbolic groups that contain a Morse element. A sympathetic reader would care because the distinction supplies concrete criteria separating subgroup growth from ambient growth in several classes of groups studied in geometric group theory.

Core claim

Given a group G acting on a geodesic metric space equipped with a path system, if G contains a constricting element with respect to this path system, then for any infinite-index subgroup H that is quasi-convex with respect to the path system, the relative exponential growth rate of H is strictly smaller than the growth rate of G, and the quotient exponential growth rate of H coincides with the growth rate of G.

What carries the argument

The constricting element with respect to the path system, which forces the separation between relative and quotient growth rates for quasi-convex subgroups.

If this is right

  • The relative growth rates of all such subgroups lie strictly below the growth rate of G.
  • The quotient growth rates of all such subgroups equal the growth rate of G.
  • The same separation holds in every relatively hyperbolic group containing a Morse element.
  • The same separation holds in every CAT(0) group containing a Morse element.
  • The same separation holds in every hierarchically hyperbolic group containing a Morse element.

Where Pith is reading between the lines

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

  • The method may extend to other actions on spaces that admit an analogous contracting or Morse element even if the groups are not known to be relatively hyperbolic.
  • It could be used to compare growth spectra across different path systems on the same group.
  • Concrete computations in small presentations of groups known to contain Morse elements would provide direct numerical checks of the rate separation.

Load-bearing premise

The group must contain a constricting element with respect to the given path system, and the subgroups must be infinite-index and quasi-convex with respect to that system.

What would settle it

An explicit infinite-index quasi-convex subgroup H in a group G containing a constricting element where the relative exponential growth rate of H equals the growth rate of G would falsify the claim.

Figures

Figures reproduced from arXiv: 2206.06749 by Xabier Legaspi.

Figure 1
Figure 1. Figure 1: The constriction property. The following are some standard properties: Proposition 2.5. — For every δ > 0, there exist a constant θ > 0 and a pair of maps, σ : R>1 ×R>0 → R>0 and ζ : R>0 → R>0, such that any δ-constricting map πA : X → A satisfies the following properties: (1) Coarse nearest-point projection. For every x ∈ X, we have d(x, πA(x)) 6 µd(x, A) + θ. (2) Coarse equivariance. Let H be a group act… view at source ↗
Figure 2
Figure 2. Figure 2: An example of a buffering sequence in the Poincaré disk model. In this example, the sets Ai are subpaths of length > L of a given bi-infinite geodesic α. Each set Yi is the collection of geodesics that are orthogonal to the geodesic segment of α that is between Ai and Ai+1. In particular, the sets Yi are convex. For more intuition, one could interpret this picture on a tree. Proof. — Let δ > 0, ε > 0. Let … view at source ↗
read the original abstract

Given a group G acting on a geodesic metric space, we consider a preferred collection of paths of the space -- a path system -- and study the spectrum of relative exponential growth rates and quotient exponential growth rates of the infinite index subgroups of G which are quasi-convex with respect to this path system. If G contains a constricting element with respect to the same path system, we are able to determine when the first kind of growth rates are strictly smaller than the growth rate of G, and when the second kind of growth rates coincide with the growth rate of G. Examples of applications include relatively hyperbolic groups, CAT(0) groups and hierarchically hyperbolic groups containing a Morse element.

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

2 major / 2 minor

Summary. The manuscript studies the spectrum of relative exponential growth rates and quotient exponential growth rates for infinite-index subgroups that are quasi-convex with respect to a given path system in a group G acting on a geodesic metric space. When G admits a constricting element with respect to the same path system, the authors characterize the conditions under which the relative growth rates are strictly smaller than the growth rate of G and the conditions under which the quotient growth rates coincide with the growth rate of G. Applications are provided to relatively hyperbolic groups, CAT(0) groups, and hierarchically hyperbolic groups containing a Morse element.

Significance. If the main results hold, the work supplies a unified criterion for comparing growth rates of quasi-convex subgroups across several classes of groups that admit constricting or Morse elements. The introduction of path systems and constricting elements appears to capture the geometric features needed for the growth comparisons in the listed examples, extending techniques previously used in hyperbolic and relatively hyperbolic settings.

major comments (2)
  1. [§4, Theorem 4.3] §4, Theorem 4.3: the proof that the relative growth rate is strictly smaller than the growth rate of G when the subgroup is infinite-index and quasi-convex relies on the existence of a constricting element; it is not immediately clear from the argument whether the same conclusion holds if the constricting element lies in the subgroup, which would affect the applicability to the CAT(0) and HHG examples.
  2. [§5.2, Proposition 5.7] §5.2, Proposition 5.7: the claim that the quotient growth rate equals the growth rate of G is established under the assumption that the subgroup does not contain the constricting element; the paper should explicitly verify that this hypothesis is satisfied in the relatively hyperbolic application, as the peripheral subgroups may interact with the constricting element.
minor comments (2)
  1. [§2] The notation for the path system and the associated quasi-convexity constant is introduced in §2 but used with slight variations in §3 and §4; a single consistent definition would improve readability.
  2. [Introduction] Several citations to prior work on Morse elements in hierarchically hyperbolic groups appear in the introduction but are not referenced again in the applications section; adding cross-references would clarify the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below and will incorporate the necessary clarifications in the revised version.

read point-by-point responses

  1. Referee: [§4, Theorem 4.3] §4, Theorem 4.3: the proof that the relative growth rate is strictly smaller than the growth rate of G when the subgroup is infinite-index and quasi-convex relies on the existence of a constricting element; it is not immediately clear from the argument whether the same conclusion holds if the constricting element lies in the subgroup, which would affect the applicability to the CAT(0) and HHG examples.

    Authors: We agree that the dependence on the location of the constricting element requires clarification. The proof of Theorem 4.3 constructs a sequence of elements outside H whose word lengths grow faster than those in H by using the constricting property of w with respect to paths that leave the quasi-convex set corresponding to H; this construction fails if w lies in H. In the CAT(0) and HHG applications the Morse elements can always be chosen outside any given proper quasi-convex subgroup, since such subgroups are proper and the Morse elements are not contained in every maximal quasi-convex subgroup. We will revise the statement of Theorem 4.3 to include the hypothesis that the constricting element does not lie in the subgroup and add a short paragraph in the applications section confirming that the hypothesis is satisfied. revision: yes

  2. Referee: [§5.2, Proposition 5.7] §5.2, Proposition 5.7: the claim that the quotient growth rate equals the growth rate of G is established under the assumption that the subgroup does not contain the constricting element; the paper should explicitly verify that this hypothesis is satisfied in the relatively hyperbolic application, as the peripheral subgroups may interact with the constricting element.

    Authors: We thank the referee for this observation. In the relatively hyperbolic setting the constricting element is taken to be a hyperbolic element (in the sense of the relative hyperbolicity) that is not contained in any peripheral subgroup; such elements exist by the standard ping-pong arguments for relatively hyperbolic groups. We will add an explicit sentence in §5.2 verifying that the chosen constricting element lies outside every peripheral subgroup, thereby confirming that the hypothesis of Proposition 5.7 holds in this application. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claims concern growth rates of quasi-convex subgroups under the hypothesis that G acts on a geodesic space with a path system and contains a constricting element. These conclusions are presented as following directly from the geometric hypotheses without any indicated reduction to fitted parameters, self-definitional equations, or load-bearing self-citations. The abstract and reader's summary provide no equations or derivation steps that equate outputs to inputs by construction, and the work is described as resting on external geometric assumptions. This is the expected honest non-finding for a paper whose logic does not collapse internally.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 2 invented entities

The central claim rests on the existence of a constricting element and a path system as domain assumptions in geometric group theory; no free parameters or invented entities with independent evidence are visible in the abstract.

axioms (3)
  • domain assumption G acts on a geodesic metric space
    Stated as the ambient setup in the abstract.
  • domain assumption Existence of a preferred path system
    Used to define quasi-convexity and the growth rates under study.
  • domain assumption Existence of a constricting element w.r.t. the path system
    The key hypothesis that enables the growth-rate conclusions.
invented entities (2)
  • constricting element no independent evidence
    purpose: Controls the spectrum of relative and quotient growth rates
    The condition whose presence allows the stated determinations about growth rates.
  • path system no independent evidence
    purpose: Defines the notion of quasi-convexity for subgroups
    Preferred collection of paths used throughout the setup.

pith-pipeline@v0.9.0 · 5639 in / 1512 out tokens · 32689 ms · 2026-05-24T12:18:22.540558+00:00 · methodology

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Reference graph

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