Rel-Cyclics Dehn Function in Group Theory

• The rel-cyclics Dehn function is a filling invariant that refines the ordinary Dehn function by measuring both the van Kampen area and the minimal exponent needed in power relations, w = u^p.
• It quantifies the complexity of conjugacy problems in fibre products by capturing the cyclic subgroup geometry inherent to torsion-free hyperbolic groups.
• Comparison theorems show that under uniform quasigeodesic conditions, the rel-cyclics Dehn function controls conjugator length functions and relates closely to classical isoperimetric bounds.

Searching arXiv for the rel-cyclics Dehn function and closely related Dehn-function literature to ground the article in current papers. The rel-cyclics Dehn function of a finitely presented group $Q$, denoted $\delta_Q^c$, is a filling invariant designed to measure the complexity of proving that one word represents a power of another, rather than merely proving triviality. It refines the ordinary Dehn function by encoding both the van Kampen area of a null-homotopic word of the form $wu^p$ and the size of the exponent $p$, and it was introduced to capture the additional cyclic-subgroup geometry that arises in the conjugacy problem for fibre products of torsion-free hyperbolic groups (Bridson, 23 Jul 2025).

1. Definition and basic variants

Let $Q=\langle X\mid R\rangle$ be a finitely presented group. If a word $w\in F(X)$ represents the identity in $Q$, one defines ${\rm Area}(w)$ as the least $M$ such that

$w=\prod_{i=1}^M \theta_i^{-1}r_i\theta_i$

in the free group, with $\delta_Q^c$0. The ordinary Dehn function is then

$\delta_Q^c$1

The rel-cyclics Dehn function is defined by

$\delta_Q^c$2

where $\delta_Q^c$3 is the order in $\delta_Q^c$4 of the element represented by $\delta_Q^c$5. The condition $\delta_Q^c$6 is equivalent to requiring that $\delta_Q^c$7 be the least non-negative integer such that $\delta_Q^c$8 in $\delta_Q^c$9. In this sense, $wu^p$0 measures the complexity of certifying that $wu^p$1 is the shortest possible power of $wu^p$2 among all pairs with $wu^p$3 (Bridson, 23 Jul 2025).

Two companion invariants are defined in the same framework. The rel-$wu^p$4-cyclics Dehn function

$wu^p$5

restricts to infinite-order elements. The strengthened variant

$wu^p$6

allows exponents up to the full order in the torsion case. In torsion-free groups one has

$wu^p$7

Like ordinary Dehn functions, these invariants are considered only up to coarse equivalence. The paper uses

$wu^p$8

and $wu^p$9 for mutual domination. It states that $p$0, $p$1, and $p$2 are well-defined up to $p$3 under change of finite presentation; similarly, the definition is unchanged up to $p$4 if one replaces $p$5 by $p$6 (Bridson, 23 Jul 2025).

2. Origin in fibre products and conjugacy

The invariant arises from the geometry of conjugacy in fibre products. Consider a short exact sequence

$p$7

with $p$8 torsion-free hyperbolic and $p$9 finitely presented, and let

$Q=\langle X\mid R\rangle$0

be the associated fibre product. Conjugacy in $Q=\langle X\mid R\rangle$1 is coordinatewise, and centralizers in torsion-free hyperbolic groups are cyclic. Consequently, after normalizing a coordinatewise conjugator, the condition that it lie in $Q=\langle X\mid R\rangle$2 becomes an equation in $Q=\langle X\mid R\rangle$3 of the form

$Q=\langle X\mid R\rangle$4

with $Q=\langle X\mid R\rangle$5 short and $Q=\langle X\mid R\rangle$6 chosen minimally modulo the order of $Q=\langle X\mid R\rangle$7. To bound the length of an actual conjugator in $Q=\langle X\mid R\rangle$8, one must both certify

$Q=\langle X\mid R\rangle$9

in $w\in F(X)$0 and control the size of $w\in F(X)$1. The quantity

$w\in F(X)$2

is exactly what $w\in F(X)$3 measures (Bridson, 23 Jul 2025).

This viewpoint explains why the ordinary Dehn function is insufficient. The usual $w\in F(X)$4 detects only null-homotopies $w\in F(X)$5. In fibre products, the decisive equations are power relations $w\in F(X)$6, because the cyclic structure of centralizers introduces an exponent parameter that must be tracked quantitatively. The rel-cyclics Dehn function is therefore not an auxiliary refinement but the invariant naturally matched to this conjugacy problem.

The same framework also yields an algorithmic interpretation. In the formulation stated in the paper, for finitely generated subdirect products $w\in F(X)$7 of torsion-free hyperbolic groups with quotient $w\in F(X)$8, the following are equivalent: the conjugacy problem in $w\in F(X)$9 is solvable; there is a uniform algorithm to decide membership of cyclic subgroups in $Q$0; and $Q$1 is recursive. This makes $Q$2 the quantitative invariant corresponding to conjugacy solvability in this class of fibre products (Bridson, 23 Jul 2025).

3. Quantitative properties and comparison inequalities

The main quantitative theorem places $Q$3 between the ordinary Dehn function of $Q$4 and conjugator length functions of the fibre product $Q$5. If $Q$6 is an epimorphism from a torsion-free hyperbolic group and $Q$7 is the associated fibre product, then

$Q$8

Thus $Q$9 controls relative conjugator length in the ambient product, while ${\rm Area}(w)$0 remains a universal lower bound (Bridson, 23 Jul 2025).

A complementary lower-bound statement first involves the infinite-order variant: ${\rm Area}(w)$1 and hence

${\rm Area}(w)$2

When ${\rm Area}(w)$3 is torsion-free, ${\rm Area}(w)$4, so one obtains

${\rm Area}(w)$5

In this regime, ${\rm Area}(w)$6 and ambient conjugator length differ by at most a linear factor.

The paper also isolates a class of groups where this comparison sharpens. If ${\rm Area}(w)$7 has uniformly quasigeodesic cyclics (UQC), then

${\rm Area}(w)$8

and if moreover ${\rm Area}(w)$9, then

$M$0

Under the same UQC hypothesis one has the corollary

$M$1

Accordingly, in the UQC setting the rel-cyclics Dehn function is, up to coarse equivalence, the correct filling invariant governing relative conjugator length.

The paper does not provide many exact closed-form computations of $M$2. Its emphasis is structural: upper and lower comparisons, equivalence theorems, and transfer principles from cyclic-subgroup geometry to conjugacy complexity.

4. Cyclic subgroup geometry and auxiliary invariants

The definition of $M$3 is built to see more than ordinary filling area. Two auxiliary functions make this explicit. The return of cyclics function is

$M$4

and the torsion-evolution function is

$M$5

The first records how large a power of an infinite-order element can return to the $M$6-ball, while the second records the largest torsion orders visible at scale $M$7 (Bridson, 23 Jul 2025).

The rel-cyclics Dehn function controls the first phenomenon but is deliberately not designed to control the second. This is why the definition uses $M$8 rather than $M$9: it isolates the minimal representative modulo the cyclic subgroup instead of folding torsion-order growth into the invariant itself. The strengthened variant $w=\prod_{i=1}^M \theta_i^{-1}r_i\theta_i$0 is the one that incorporates the full torsion range.

Under UQC, the exponent $w=\prod_{i=1}^M \theta_i^{-1}r_i\theta_i$1 is automatically linearly controlled: if $w=\prod_{i=1}^M \theta_i^{-1}r_i\theta_i$2, then $w=\prod_{i=1}^M \theta_i^{-1}r_i\theta_i$3 whenever $w=\prod_{i=1}^M \theta_i^{-1}r_i\theta_i$4. This is the mechanism behind the estimate $w=\prod_{i=1}^M \theta_i^{-1}r_i\theta_i$5. The paper also discusses uniform monotonicity of cyclic subgroups, a stronger property than UQC, and notes that it may sharpen area estimates for words $w=\prod_{i=1}^M \theta_i^{-1}r_i\theta_i$6.

An instructive counterpoint is the example

$w=\prod_{i=1}^M \theta_i^{-1}r_i\theta_i$7

which has UQC but not uniformly monotone cyclics. This separates two kinds of cyclic-subgroup geometry that might otherwise be conflated. A second illustrative remark concerns $w=\prod_{i=1}^M \theta_i^{-1}r_i\theta_i$8: although

$w=\prod_{i=1}^M \theta_i^{-1}r_i\theta_i$9

the paper notes that if a word $\delta_Q^c$00 of length $\delta_Q^c$01 traces a path staying inside a ball of radius $\delta_Q^c$02, then

$\delta_Q^c$03

This suggests that group-specific estimates for $\delta_Q^c$04 can be substantially sharper than the generic $\delta_Q^c$05 bound (Bridson, 23 Jul 2025).

5. Relation to other Dehn-function notions

The rel-cyclics Dehn function is distinct from several nearby invariants with similar names. It is not the relative Dehn function of a pair $\delta_Q^c$06 in the sense of finite relative presentations. That theory studies null-homotopies relative to designated peripheral subgroups and is developed, for example, via fine Cayley-Abels graphs and combination theorems for graphs of groups (Bigdely et al., 2022). By contrast, $\delta_Q^c$07 is attached to a single finitely presented group $\delta_Q^c$08 and measures the difficulty of proving equations $\delta_Q^c$09.

It is also different from the Dehn functions of cocyclic or coabelian kernels, such as kernels of maps from direct products onto $\delta_Q^c$10 or $\delta_Q^c$11. That literature studies ordinary Dehn functions of subgroups like

$\delta_Q^c$12

with exact or near-exact asymptotics in terms of the ambient direct product (Kropholler et al., 2021). The rel-cyclics Dehn function, by contrast, is not a subgroup Dehn function at all; it is a power-membership-sensitive filling invariant in the quotient group $\delta_Q^c$13.

A further adjacent theme is the Dehn-function theory of free-by-cyclic groups and their doubles, where cyclic extensions produce large ordinary Dehn functions or large Dehn functions in finitely presented subgroups of right-angled Artin groups (Brady et al., 2017). Those results concern standard isoperimetric functions. The rel-cyclics Dehn function instead enters when cyclic centralizers in hyperbolic groups force one to solve quantitative power equations in a quotient.

A common misconception is therefore to read “rel-cyclics” as “relative to cyclic subgroups” in the sense of relatively hyperbolic geometry or relative presentations. The invariant introduced in (Bridson, 23 Jul 2025) is more specialized: it is a Dehn-type function for minimal power certification.

6. Classes of groups, realizations, and significance

The paper identifies several classes of groups where $\delta_Q^c$14 is estimable. Groups with UQC include torsion-free hyperbolic groups, groups acting freely, properly, and cocompactly on CAT(0) spaces, finitely generated subgroups of such groups, certain torsion-free subgroups of $\delta_Q^c$15, and some products and HNN extensions. For any finitely presented group with UQC,

$\delta_Q^c$16

so in these settings the rel-cyclics Dehn function is bounded by at most a quadratic reparameterization of the ordinary Dehn function (Bridson, 23 Jul 2025).

The paper also uses $\delta_Q^c$17 as a construction tool. From any finitely presented $\delta_Q^c$18, it builds a quotient $\delta_Q^c$19 with UQC such that

$\delta_Q^c$20

Fibre products over $\delta_Q^c$21 then produce finitely presented groups whose conjugator length functions are controlled via $\delta_Q^c$22. This provides a route from ordinary Dehn-function growth in quotients to large conjugator length functions in finitely presented groups.

The invariant is therefore significant on three levels. Formally, it extends the classical Dehn function to a power-sensitive setting. Geometrically, it encodes cyclic-subgroup distortion data that the ordinary Dehn function does not detect. Algorithmically, within the fibre-product framework of torsion-free hyperbolic groups, it is the exact coarse invariant corresponding to solvability of conjugacy.

Its present limitations are equally clear. The theory is strongest in comparison theorems and coarse equivalences, not in exact computations for broad natural classes. The paper itself emphasizes estimates and structural applications rather than a complete catalogue of explicit $\delta_Q^c$23-values. Even so, it establishes the rel-cyclics Dehn function as the natural filling invariant for conjugacy phenomena governed by cyclic centralizers and minimal exponent equations (Bridson, 23 Jul 2025).