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

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.

• The paper proves that FP2 kernels in free extensions have polynomial homological Dehn functions controlled by area-radius bounds in the ambient group.
• It adapts classical Van Kampen diagram techniques into a novel framework using surface diagrams, facilitating the analysis of non-finitely presented groups.
• The work extends Gersten–Short methods by providing explicit polynomial bounds, advancing quantitative insights in geometric group theory.

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

Introduction and Motivation

The paper "Homological Isoperimetric Inequalities for Kernels of Free Extensions of Type $FP_2$" (2604.04549) addresses a fundamental question in geometric group theory: the behavior of (homological) isoperimetric functions for groups arising as kernels in extensions with free quotients. For hyperbolic groups, classic results tie hyperbolicity to the existence of a linear isoperimetric (Dehn) function, but the situation for subgroups, particularly those not finitely presented, is significantly more intricate.

Notably, the work extends results analogous to Gersten and Short [Gersten & Short, 2002] to the context of homological Dehn functions, providing homological isoperimetric inequalities for kernels $K$ of extensions

$1 \to K \to H \to F_n \to 1$

where $F_n$ is free and $K$ has type $FP_2$. This is achieved even when $K$ is not finitely presentable.

Homological Dehn Functions: Context and Definitions

The homological Dehn function $\FA_G$ provides a measure of the "filling area" of loops in terms of 2-chains in a suitable complex on which $G$ acts. While classically defined for finitely presented groups (where it agrees up to equivalence with the Dehn function, a measure with strong algorithmic implications), $FP_2$0 can be defined for groups of type $FP_2$1, broadening the class of interest to groups not necessarily finitely presented (cf. Bestvina-Brady [bestvina1997, (Brady et al., 2020)]).

A group $FP_2$2 is of type $FP_2$3 if it admits a projective resolution of the trivial module by finitely generated projectives up to dimension $FP_2$4, or equivalently, has a finite 1-skeleton and finitely many 2-relators making the relevant $FP_2$5 vanish (homological finite presentation). The paper provides rigorous constructions in this more general setting and adapts the combinatorial tools of Van Kampen diagrams to "surface diagrams."

Main Technical Results

The core contributions are:

1. Extension of Isoperimetric Bounds: The paper proves that if $FP_2$6 is of type $FP_2$7 and is a free extension of $FP_2$8 of type $FP_2$9, then the homological Dehn function of $K$0 is controlled in terms of a homological area-radius pair $K$1 for $K$2:

   $K$3

   for some constant $K$4, where $K$5 are the area and radius bounds for fillings in $K$6.

2. Hyperbolicity Implications: For $K$7 hyperbolic, the classical results ensure a polynomial (often linear or near-linear) isoperimetric function for $K$8. The extension then yields,

   $K$9

   Hence, kernels in such extensions, even when not finitely presentable, satisfy polynomial homological isoperimetric inequalities.

3. Combinatorial Frameworks: The adaptation of area-radius pairs to the homological context leverages "surface diagrams" generalizing Van Kampen diagrams, making the arguments valid for arbitrary $1 \to K \to H \to F_n \to 1$0 groups.
4. Generalization of Gersten-Short Techniques: The process of "pushing down" $1 \to K \to H \to F_n \to 1$1-rings in disc diagrams is replaced with analysis on $1 \to K \to H \to F_n \to 1$2-cycles in the generality of surface diagrams. The proof handles new complications (e.g., $1 \to K \to H \to F_n \to 1$3-cycles not being nullhomotopic) via a detailed study of the group's Cayley complex structure and the coarse geometry of the extension.

Numerical and Structural Strength

Theorems in the paper provide explicit polynomial upper bounds for the homological Dehn function of $1 \to K \to H \to F_n \to 1$4 when $1 \to K \to H \to F_n \to 1$5 is hyperbolic. These are derived by propagating linear and logarithmic area and radius bounds in $1 \to K \to H \to F_n \to 1$6 through the extension mechanism. The argument critically relies on the control of area growth under automorphisms induced by conjugation by stable letters, with precise constants determined in Lemmas analogous to those in [gersten2002].

The analysis provides both existential results (the bounds exist) and constructive bounds (the structure of surface diagrams and area-radius does not explode super-polynomially) for a large class of subgroups previously inaccessible to isoperimetric analysis.

Implications and Theoretical Impact

Theoretical Significance

• Broader Class of Groups: The results connect the homological isoperimetric theory to the field of groups not finitely presented but still accessible via homological finiteness ($1 \to K \to H \to F_n \to 1$7). This includes "Bestvina-Brady" groups, subgroups constructed via Rips-type or branched covering techniques, and extensions where pathological behaviors (e.g., unsolvable word problem) may occur, but with controlled filling functions [bestvina1997, brady1999, kropholler2021hyperbolicgroupsfinitelypresented].
• Homological vs. Homotopical Dehn Functions: The techniques and results distinguish the subtleties between homological and homotopical Dehn functions. It is emphasized that while algorithmic properties (e.g., solvability of the word problem) may not transfer, quantitative isoperimetric and filling properties can still exhibit strong control in the homological setting (Brady et al., 2020).

Practical and Methodological Ramifications

• Applications in Cohomological Dimension and Group Construction: The explicit control over subgroups' Dehn functions is instrumental for constructions in geometric group theory and topology, especially given recent constructions of hyperbolic groups with subgroups of prescribed finiteness properties [kropholler2021hyperbolicgroupsfinitelypresented, isenrich2024].
• Foundations for Further Quantitative Analysis: The area-radius pair framework in the homological context offers a pathway for effective bounds on higher-order filling invariants and may influence future work on $1 \to K \to H \to F_n \to 1$8 properties or systematic control over isoperimetric spectra [Brady2000, (Kropholler et al., 3 Jan 2025)].

Speculation on Future Directions

The present methodology—relating area-radius pairs in a group to those in subgroups via free extensions—suggests avenues for:

• Similar Analysis in Non-free Extensions: Extension of these results to more general quotients (e.g., abelian, nilpotent, or even more general automatic groups) could further generalize the class of accessible kernels.
• Automata-theoretic and Logic Connections: Since algorithmic finiteness and isoperimetric inequality properties diverge in the homological context, the precise mapping between algebraic, geometric, and decision-theoretic characteristics remains an open and rich field.

Conclusion

The paper establishes homological isoperimetric inequalities for kernels of free extensions in type $1 \to K \to H \to F_n \to 1$9 groups, showing that the homological Dehn function of such kernels is controlled—often polynomially—for a broad swath of groups including those not finitely presentable. By extending the Gersten-Short paradigm to a homological framework, employing combinatorial constructs of surface diagrams, and controlling area and radius growth under automorphisms and extensions, the work enriches both the quantitative and qualitative understanding of isoperimetric phenomena in geometric group theory. These results not only close a significant gap in the literature but also provide a robust foundation for further exploration of finiteness and filling invariants in complex group-theoretic constructions.