Homological Isoperimetric Inequalities for Kernels of Free Extensions of Type $FP_2$

Jakub F. Tucker

arxiv: 2604.04549 · v2 · submitted 2026-04-06 · 🧮 math.GR

Homological Isoperimetric Inequalities for Kernels of Free Extensions of Type FP₂

Jakub F. Tucker This is my paper

Pith reviewed 2026-05-10 19:56 UTC · model grok-4.3

classification 🧮 math.GR

keywords homological isoperimetric inequalityFP₂ subgroupsfree extensionskernelssurface diagramsgroup homologyisoperimetric functions

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The pith

Kernels of free extensions of type FP₂ satisfy homological isoperimetric inequalities.

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

The paper defines homological area-radius pairs using surface diagrams. It adapts the Gersten-Short argument to prove that subgroups of type FP₂ arising as kernels of free extensions obey a homological isoperimetric inequality. This inequality bounds the homological area of cycles in terms of their radii. A reader would care because such bounds often constrain the geometric and homological complexity of groups, giving control over fillings and presentations in this specific class.

Core claim

The central claim is that by equipping the setting with homological area-radius pairs defined via surface diagrams, the Gersten-Short proof carries over to yield a homological isoperimetric inequality for any subgroup H of type FP₂ that appears as the kernel of a free extension.

What carries the argument

Homological area-radius pairs with surface diagrams, which associate homological areas to cycles and radii measured on surface diagrams so that the Gersten-Short filling argument applies directly.

Load-bearing premise

The newly defined homological area-radius pairs with surface diagrams are sufficiently well-behaved that the Gersten-Short argument applies verbatim to kernels of free extensions of type FP₂.

What would settle it

Construct an explicit free extension whose kernel is of type FP₂ but for which some cycle has homological area that grows faster than any function of its radius under the defined pairs.

Figures

Figures reproduced from arXiv: 2604.04549 by Jakub F. Tucker.

**Figure 1.** Figure 1: Gluing edges around h, highlighted edges may be glued with the orientation shown. are no unglued edges. As in [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

**Figure 2.** Figure 2: Near h is a surface (with boundary). If this process appears to terminate early, that is, we have a closed surface with boundary S, but there are still 2-cells in c unaccounted for in S, then these comprise a separate connected component in S. We can continue to build our surface diagram by picking any unaccounted-for 2-cell in c and repeating the construction. Since the 2-chain c is finite, we have only f… view at source ↗

**Figure 3.** Figure 3: Obtaining the surface diagram S ′ by refining a surface diagram S. Let P0 = ⟨A∥R0⟩,P1 = ⟨A∥R1⟩ be homological finite presentations for H. Consider the homological Cayley complex X0 for P0. Note that this space shares its 1-skeleton with the homological Cayley complex X1 for P1: both are the Cayley graph for H with respect to generating set A. Let γ be a loop of length at most n in X (1) 0 with surface diag… view at source ↗

**Figure 4.** Figure 4: An internal A-edge. □ 4.3. A Homological Finite Presentation for H. Now we are able to state an analogue of Gersten and Short’s result for groups of type F P2: Theorem A. Let H be an extension of a group K of type F P2 by a finitely generated free group Fn, so that we have the short exact sequence 1 → K → H → Fn → 1. Then H is of type F P2, and if (f, g) is a homological area-radius pair for H with f(n) ≥ … view at source ↗

**Figure 5.** Figure 5: Sketch of part of Γ(H, A ∪ {t}), where n = 1. We construct a 2-complex X from Γ(H, A ∪ {ti , . . . , tn}) by gluing, for each vertex x ∈ H and for each of the finitely many relations r ∈ R0 ∪ {t −1 i aj tiΦi(aj ) −1 ∣ aj ∈ A, i ∈ {1, . . . , n}}, a 2-cell with boundary the loop labelled by r beginning at x. Under this construction, the copies of Γ(K, A) corresponding to cosets Kw in H have become copies of… view at source ↗

read the original abstract

We define homological area-radius pairs with surface diagrams. Using these, we adapt a proof of Gersten and Short \cite{gersten2002} to obtain a homological isoperimetric inequality for subgroups of type $FP_2$ which appear as kernels of free extensions.

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.

Desk Editor's Note private letter to a colleague

Adapts Gersten-Short via new homological area-radius pairs to get an isoperimetric inequality for FP2 kernels of free extensions.

read the letter

This paper defines homological area-radius pairs with surface diagrams and adapts the Gersten-Short 2002 argument to produce a homological isoperimetric inequality for kernels of free extensions that have type FP2. That is the actual new piece: a concrete filling bound for this restricted but natural class of subgroups that had not been handled before in the homological setting. The author keeps the work focused and cites the source result appropriately without claiming more than the adaptation delivers. The setup looks clean on the surface and gives a usable statement for people who need bounds on these particular subgroups. The soft spot is exactly where the stress-test note flags it. The whole argument rests on the new pairs satisfying the same subadditivity, monotonicity, and diagram-filling relations that the original combinatorial proof uses. If surface diagrams bound homological area in a way that differs at the critical steps, the transfer does not go through verbatim. The abstract presents the adaptation as successful, so the manuscript must contain the verification, but a referee will want to see those checks spelled out rather than assumed. This is a technical note aimed at geometric group theorists who already know the Gersten-Short work and care about homological finiteness conditions. It will not reorganize the field, but a reader in that niche can extract the bound and the new definition without much extra effort. The paper is coherent on its own terms and shows honest engagement with the literature, so it deserves a serious referee rather than a desk rejection. A review can confirm the property transfer and suggest any tightening of the new objects.

Referee Report

1 major / 0 minor

Summary. The paper defines homological area-radius pairs with surface diagrams and adapts a proof of Gersten and Short to obtain a homological isoperimetric inequality for subgroups of type FP₂ which appear as kernels of free extensions.

Significance. If valid, the result extends isoperimetric inequalities into the homological setting for FP₂ kernels of free extensions, offering potential new tools for analyzing filling functions and subgroup properties in geometric group theory. The adaptation of an established external proof is a methodological strength provided the key relations transfer without mismatch.

major comments (1)

The adaptation of the Gersten-Short argument relies on the newly defined homological area-radius pairs (with surface diagrams) inheriting the precise subadditivity, monotonicity, and diagram-filling relations from the combinatorial case. The manuscript must explicitly verify these properties for the homological versions, as any discrepancy in how surface diagrams bound homological area versus combinatorial area would break the argument at the step where the original proof invokes those relations. This is the load-bearing step for the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the major comment below.

read point-by-point responses

Referee: The adaptation of the Gersten-Short argument relies on the newly defined homological area-radius pairs (with surface diagrams) inheriting the precise subadditivity, monotonicity, and diagram-filling relations from the combinatorial case. The manuscript must explicitly verify these properties for the homological versions, as any discrepancy in how surface diagrams bound homological area versus combinatorial area would break the argument at the step where the original proof invokes those relations. This is the load-bearing step for the central claim.

Authors: We agree this verification is essential for the adaptation to be rigorous. The definitions of homological area-radius pairs via surface diagrams are constructed so that homological area is bounded in direct analogy to combinatorial area, which ensures the required subadditivity, monotonicity, and diagram-filling relations hold by the same combinatorial arguments applied to the underlying surface diagrams. Nevertheless, to make this transfer fully explicit as requested, the revised manuscript will include a new subsection that states and proves these three properties for the homological setting, confirming there is no mismatch with the combinatorial case and that the Gersten-Short argument applies without modification. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation adapts external Gersten-Short proof via new definitions

full rationale

The paper defines homological area-radius pairs with surface diagrams and then adapts the 2002 Gersten-Short argument (external citation) to obtain the isoperimetric inequality for FP₂ kernels. No self-citations appear, no parameters are fitted to data and relabeled as predictions, and no equations reduce by construction to the inputs. The load-bearing step is the transfer of filling/radius properties to the new homological pairs, but this is a question of correctness of the adaptation rather than circularity. The derivation remains self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces one new definition (homological area-radius pairs) whose validity is taken as a domain assumption; no free parameters or invented entities are mentioned.

axioms (1)

domain assumption Homological area-radius pairs with surface diagrams can be defined so that the Gersten-Short argument applies directly to the kernels in question.
This assumption is required for the adaptation step described in the abstract.

pith-pipeline@v0.9.0 · 5329 in / 1221 out tokens · 51067 ms · 2026-05-10T19:56:04.048787+00:00 · methodology

discussion (0)

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Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

[AW22] Macarena Arenas and Daniel T. Wise. Linear isoperimetric functions for surfaces in hyperbolic groups, 2022. [BB97] Mladen Bestvina and Noel Brady. Morse theory and finiteness properties of groups.Inventiones mathe- maticae, 129(3):445–470, Aug 1997. HOMOLOGICAL ISOPERIMETRIC INEQUALITIES FOR KERNELS OF FREE EXTENSIONS OF TYPEF P 2 18 [BKS21] Noel B...

work page 2022

[1] [1]

[AW22] Macarena Arenas and Daniel T. Wise. Linear isoperimetric functions for surfaces in hyperbolic groups, 2022. [BB97] Mladen Bestvina and Noel Brady. Morse theory and finiteness properties of groups.Inventiones mathe- maticae, 129(3):445–470, Aug 1997. HOMOLOGICAL ISOPERIMETRIC INEQUALITIES FOR KERNELS OF FREE EXTENSIONS OF TYPEF P 2 18 [BKS21] Noel B...

work page 2022