Conjugator Length Function in Groups

Updated 7 July 2026

Conjugator length function is a quantitative invariant that measures the minimal length of a conjugator required to witness conjugacy between pairs with bounded word length.
It captures group geometry and complexity by linking subgroup distortion, quotient metrics, and centralizer structures to the cost of establishing conjugacy.
Different classes of groups, from hyperbolic to nilpotent, display distinct conjugator length growth regimes, ranging from linear bounds to non-elementary behaviors.

Searching arXiv for recent and foundational papers on the conjugator length function. The conjugator length function is a quantitative invariant of the conjugacy problem in a finitely generated group. Instead of asking only whether two elements are conjugate, it asks how long a shortest conjugator must be in the worst case among conjugate pairs of bounded total length. In the standard formulation, for a finitely generated group $\Gamma$ with word length $|\cdot|$ , one sets

$CLF_\Gamma(n)=\max \left\{ \min \{ |w| : wu=vw \} : |u| + |v| \leq n \text{ and $u $is conjugate to$ v$ in }\Gamma \right\},$

so $CLF_\Gamma(n)$ measures the maximal minimal length of a conjugator needed to witness conjugacy at scale $n$ (Sale, 2013). This invariant is repeatedly presented as an “effective” refinement of the conjugacy problem, analogous to the way the Dehn function refines the word problem, and it has become a central tool for comparing conjugacy complexity across geometric, nilpotent, solvable, hyperbolic, and finitely presented groups (Sale, 2013).

1. Definition and basic interpretation

In the standard finitely generated setting, the conjugator length function assigns to each $n$ the least uniform bound on the length of a shortest conjugator for all conjugate pairs $u,v$ with $|u|+|v|\le n$ (Sale, 2013). Equivalent formulations emphasize the same geometric content: $CLF_\Gamma(n)$ records the length of the shortest possible conjugator in the worst conjugate pair of size at most $n$ . This makes the invariant sensitive not merely to solvability of conjugacy, but to the metric cost of exhibiting conjugacy.

Several papers make explicit that the invariant is well defined only up to the usual coarse equivalence associated with changing finite generating sets (Gillis et al., 12 Jan 2026). In the finitely presented setting this coarse viewpoint is standard: one writes $|\cdot|$ 0 if there exists $|\cdot|$ 1 such that

$|\cdot|$ 2

The same idea extends beyond discrete finitely generated groups. For a compactly generated locally compact Hausdorff group $|\cdot|$ 7 with compact generating set $|\cdot|$ 8, one defines the conjugator length

$|\cdot|$ 9

and then

$CLF_\Gamma(n)=\max \left\{ \min \{ |w| : wu=vw \} : |u| + |v| \leq n \text{ and $u $is conjugate to$ v$ in }\Gamma \right\},$0

In that setting the growth type of $CLF_\Gamma(n)=\max \left\{ \min \{ |w| : wu=vw \} : |u| + |v| \leq n \text{ and $u $is conjugate to$ v$ in }\Gamma \right\},$1 is independent of the compact generating set up to quasi-equivalence, and abelian groups are conventionally assigned constant zero conjugator length (Rego et al., 2 Jul 2025).

A substantial part of the theory concerns variants that isolate specific geometric mechanisms. In group extensions

$CLF_\Gamma(n)=\max \left\{ \min \{ |w| : wu=vw \} : |u| + |v| \leq n \text{ and $u $is conjugate to$ v$ in }\Gamma \right\},$2

the relevant auxiliary invariants are the twisted conjugacy length function and the restricted conjugacy length function. For an automorphism $CLF_\Gamma(n)=\max \left\{ \min \{ |w| : wu=vw \} : |u| + |v| \leq n \text{ and $u $is conjugate to$ v$ in }\Gamma \right\},$3, twisted conjugacy replaces the equation $CLF_\Gamma(n)=\max \left\{ \min \{ |w| : wu=vw \} : |u| + |v| \leq n \text{ and $u $is conjugate to$ v$ in }\Gamma \right\},$4 with

$CLF_\Gamma(n)=\max \left\{ \min \{ |w| : wu=vw \} : |u| + |v| \leq n \text{ and $u $is conjugate to$ v$ in }\Gamma \right\},$5

and the associated function measures how large $CLF_\Gamma(n)=\max \left\{ \min \{ |w| : wu=vw \} : |u| + |v| \leq n \text{ and $u $is conjugate to$ v$ in }\Gamma \right\},$6 must be in terms of $CLF_\Gamma(n)=\max \left\{ \min \{ |w| : wu=vw \} : |u| + |v| \leq n \text{ and $u $is conjugate to$ v$ in }\Gamma \right\},$7. The restricted version asks for conjugacy of two elements in a subgroup $CLF_\Gamma(n)=\max \left\{ \min \{ |w| : wu=vw \} : |u| + |v| \leq n \text{ and $u $is conjugate to$ v$ in }\Gamma \right\},$8 but allows the conjugator to lie in the ambient group (Sale, 2012). These refinements are not cosmetic: in extension theorems they interact with subgroup distortion and with the geometry of projected centralizers.

A different refinement is the permutation conjugacy length function $CLF_\Gamma(n)=\max \left\{ \min \{ |w| : wu=vw \} : |u| + |v| \leq n \text{ and $u $is conjugate to$ v$ in }\Gamma \right\},$9. Here one is allowed first to cyclically permute the input words. If $CLF_\Gamma(n)$ 0 and $CLF_\Gamma(n)$ 1 are cyclic permutations of $CLF_\Gamma(n)$ 2 and $CLF_\Gamma(n)$ 3, a word $CLF_\Gamma(n)$ 4 is a PC-conjugator if

$CLF_\Gamma(n)$ 5

The basic comparison

$CLF_\Gamma(n)$ 6

shows that $CLF_\Gamma(n)$ 7 can be strictly smaller while remaining asymptotically close whenever $CLF_\Gamma(n)$ 8 is sublinear or linear (Antolín et al., 2015). In hyperbolic and relatively hyperbolic settings this variant is especially effective because the conjugacy geometry is naturally encoded by cyclic shifts and thin polygon arguments.

The closest diagrammatic companion to $CLF_\Gamma(n)$ 9 is the annular Dehn function. In a finitely presented group, if $n$ 0 and $n$ 1 are conjugate, the quantity $n$ 2 can be interpreted as the minimal length of a path in an annular diagram connecting the marked boundary vertices, and

$n$ 3

captures this minimal path length uniformly (Gillis et al., 24 Jun 2025). Brick and Corson’s inequalities place $n$ 4, the annular Dehn function $n$ 5, and the ordinary Dehn function $n$ 6 into a common framework:

$n$ 7

and

$n$ 8

with $n$ 9 depending on the presentation (Gillis et al., 24 Jun 2025). These inequalities are fundamental, but later work shows they are often far from sharp.

3. Structural mechanisms behind upper bounds

One of the main themes in the subject is that upper bounds on conjugator length are usually driven by a combination of quotient geometry, subgroup distortion, and centralizer structure. In extension theory, the general theorem of “Conjugacy Length in Group Extensions” expresses $n$ 0 in terms of the quotient conjugacy length $n$ 1, the restricted conjugacy length $n$ 2, twisted conjugacy in $n$ 3, the distortion functions $n$ 4 and $n$ 5, and a centralizer-diameter parameter $n$ 6 (Sale, 2012). This formulation explains why some semidirect products $n$ 7 have linear conjugacy length while certain $n$ 8 with $n$ 9 admit only exponential upper bounds in that framework (Sale, 2012).

In wreath products the controlling mechanism is especially explicit. For $u,v$ 0, the conjugator $u,v$ 1 splits into a $u,v$ 2-component and a function component, and the paper “The Geometry of the Conjugacy Problem in Wreath Products and Free Solvable Groups” repeatedly shows that one first bounds the $u,v$ 3-part $u,v$ 4, then the whole conjugator (Sale, 2013). In the infinite-order case, if $u,v$ 5 and $u,v$ 6 are conjugate, there is a conjugator satisfying

$u,v$ 7

where $u,v$ 8, and $u,v$ 9 is either $|u|+|v|\le n$ 0 or $|u|+|v|\le n$ 1 depending on whether $|u|+|v|\le n$ 2 is conjugate to $|u|+|v|\le n$ 3 (Sale, 2013). In the finite-order case the estimate changes to

$|u|+|v|\le n$ 4

so the conjugacy length function of $|u|+|v|\le n$ 5 enters explicitly (Sale, 2013). This contrast isolates a key phenomenon: cyclic subgroup distortion in the quotient can dominate the geometry of short conjugators.

Hierarchically hyperbolic groups provide another large-scale mechanism. For any HHG $|u|+|v|\le n$ 6, conjugate Morse elements satisfy a linear estimate

$|u|+|v|\le n$ 7

for a shortest conjugator $|u|+|v|\le n$ 8, and in a subclass with $|u|+|v|\le n$ 9 stabilizers, orthogonal decomposition, and the commutative property, there exist $CLF_\Gamma(n)$ 0 and a uniform $CLF_\Gamma(n)$ 1 such that every pair of conjugate infinite-order elements satisfies

$CLF_\Gamma(n)$ 2

(Abbott et al., 2018). The result covers, in the stated sense, mapping class groups, right-angled Artin groups, compact special groups, virtually compact special groups, and related HHGs (Abbott et al., 2018).

A complementary locally compact mechanism appears for split subgroups $CLF_\Gamma(n)$ 3 of $CLF_\Gamma(n)$ 4. There one replaces word length by the norm of the translation part, reduces conjugacy to the linear equation

$CLF_\Gamma(n)$ 5

and then bounds minimal solutions via the Moore–Penrose pseudoinverse. Because $CLF_\Gamma(n)$ 6 is compact, the operator norms of $CLF_\Gamma(n)$ 7 are uniformly bounded, which yields linear growth of the conjugator length function for such groups, including $CLF_\Gamma(n)$ 8, affine Coxeter groups, and split crystallographic groups (Rego et al., 2 Jul 2025).

4. Growth regimes and representative examples

The literature now exhibits a broad range of asymptotic behaviors, from exact linear bounds to non-elementary growth.

Setting	Conjugator length behavior	Source
Lamplighter groups $CLF_\Gamma(n)$ 9	$n$ 0	(Sale, 2011)
Free-by-cyclic groups $n$ 1	$n$ 2	(Bridson et al., 2 Jun 2025)
Surface group $n$ 3, $n$ 4	$n$ 5	(Wang et al., 17 Mar 2026)
Thompson’s group $n$ 6	$n$ 7	(Belk et al., 2021)
Free solvable groups $n$ 8, $n$ 9	conjugacy length is a cubic polynomial	(Sale, 2013)
Model filiform groups $\|\cdot\|$ 00	$\|\cdot\|$ 01	(Bridson et al., 2 Jun 2025)
Certain 2-step nilpotent groups $\|\cdot\|$ 02	$\|\cdot\|$ 03	(Bridson et al., 2 Jun 2025)
Snowflake central extensions $\|\cdot\|$ 04	$\|\cdot\|$ 05, $\|\cdot\|$ 06	(Bridson et al., 16 Dec 2025)
Baumslag–Gersten group	tower of exponentials of logarithmic height	(Gillis, 29 Jul 2025)
Central extensions $\|\cdot\|$ 07 of $\|\cdot\|$ 08	$\|\cdot\|$ 09	(Bridson et al., 29 Dec 2025)

Some of these examples are sharp in a very strong sense. In lamplighter groups the bound $|\cdot|$ 10 comes from explicit control of the common-ancestor geometry in the Diestel–Leader model (Sale, 2011). For closed surface groups with the symmetric presentation, the exact parity relation

$|\cdot|$ 11

is proved together with the lower bound obtained from

$|\cdot|$ 12

whose shortest conjugator has length $|\cdot|$ 13 (Wang et al., 17 Mar 2026). For Thompson’s group $|\cdot|$ 14, the upper bound

$|\cdot|$ 15

is accompanied by explicit conjugate pairs whose shortest conjugator has length $|\cdot|$ 16, giving exact quadratic growth (Belk et al., 2021).

Polynomial behavior now occurs in arbitrarily high degrees. The model filiform groups $|\cdot|$ 17 realize $|\cdot|$ 18 for every $|\cdot|$ 19, and the family of 2-step nilpotent groups $|\cdot|$ 20 realizes $|\cdot|$ 21 (Bridson et al., 2 Jun 2025, Bridson et al., 2 Jun 2025). More generally, “Conjugator Length in Finitely Presented Groups” proves that every function realizable as the Dehn function of a finitely presented group is also realizable, up to coarse equivalence, as a conjugator length function of a finitely presented group. In particular, for every real $|\cdot|$ 22 computable in double-exponential time, there exists a finitely presented group with $|\cdot|$ 23 (Gillis et al., 12 Jan 2026). The snowflake constructions complement this by producing non-integer exponents dense in $|\cdot|$ 24 (Bridson et al., 16 Dec 2025).

At the opposite end of the spectrum lie the fastest known examples. For the Baumslag–Gersten group,

$|\cdot|$ 25

so the conjugator length function grows faster than any tower of exponentials of fixed height (Gillis, 29 Jul 2025). Fibre product constructions then push the spectrum into the Grzegorczyk hierarchy: for finitely presented groups $|\cdot|$ 26, the conjugator length function grows like functions at the $|\cdot|$ 27-th primitive-recursive level, in the precise sense

$|\cdot|$ 28

(Bridson et al., 29 Dec 2025).

5. Algorithmic and normal-form aspects

Because $|\cdot|$ 29 bounds the search space for conjugators, it has direct algorithmic meaning. The lamplighter paper states that if $|\cdot|$ 30 is finitely generated as an abelian group, then there is an algorithm deciding conjugacy in $|\cdot|$ 31 and producing a conjugator in time $|\cdot|$ 32, where $|\cdot|$ 33 is the total input length (Sale, 2011). In the free-by-cyclic groups $|\cdot|$ 34, the proof of linear conjugator length is constructive and yields polynomial-time algorithms for both the conjugacy problem and the conjugacy search problem (Bridson et al., 2 Jun 2025).

The surface-group literature illustrates two different algorithmic layers. “Word Length Formulae and Normal Forms of Conjugacy Classes in Surface Groups” gives a unique normal form $|\cdot|$ 35, a canonical representative $|\cdot|$ 36 for conjugacy classes, a conjugacy criterion, and algorithms that compute $|\cdot|$ 37 in linear time and $|\cdot|$ 38 together with a conjugator in about $|\cdot|$ 39 (Wang et al., 17 Nov 2025). At the same time, that paper explicitly does not define the asymptotic conjugator length function or prove general growth bounds for it (Wang et al., 17 Nov 2025). The later paper “An Explicit Bound for the Conjugator Length Function of a Surface Group” turns this normal-form and reduction analysis into an explicit asymptotic theorem, with the fully quantified bound

$|\cdot|$ 40

(Wang et al., 17 Mar 2026).

Permutation conjugacy length supplies a different algorithmic improvement. In relatively hyperbolic groups, the main theorem gives

$|\cdot|$ 41

so the permutation conjugacy length of the ambient group is controlled by that of the parabolic subgroups (Antolín et al., 2015). When $|\cdot|$ 42 is bounded by a computable function and the word problem has complexity $|\cdot|$ 43, the resulting conjugacy algorithm runs in time

$|\cdot|$ 44

and if $|\cdot|$ 45 is bounded by a constant $|\cdot|$ 46, this becomes

$|\cdot|$ 47

(Antolín et al., 2015). This does not replace $|\cdot|$ 48, but it refines the metric search problem by incorporating cyclic shifts.

6. Relations to filling invariants, flexibility, and open directions

A major development is the recognition that conjugator length is not determined by the ordinary Dehn function or by the annular Dehn function. “Distinguishing Filling Invariants Associated to Conjugacy in Groups” proves that these three invariants are independent up to coarse equivalence (Gillis et al., 24 Jun 2025). The examples are stark: the Heisenberg group $|\cdot|$ 49 satisfies

$|\cdot|$ 50

while $|\cdot|$ 51 satisfies

$|\cdot|$ 52

(Gillis et al., 24 Jun 2025). These examples show that annular area and shortest conjugator length can diverge sharply in either direction.

The subject also contains unresolved structural questions. In hierarchically hyperbolic groups the available linear theorems cover Morse elements in full generality and suitable powers of infinite-order elements in a large subclass, but the paper explicitly notes that this is not a full solution for all HHGs, that finite-order elements remain unresolved, and that the authors expect the conjugator length function could even be exponential in some HHGs (Abbott et al., 2018). For free solvable groups, the cubic upper bound is established, but the earlier wreath-product analysis leaves open whether in some cases the exact exponent lies strictly between $|\cdot|$ 53 and $|\cdot|$ 54, and it specifically suggests that the metabelian case $|\cdot|$ 55 may even be subquadratic (Sale, 2013).

Recent spectrum results make these open questions more striking rather than less. The spectrum of conjugator length functions of finitely presented groups contains every Dehn function spectrum value (Gillis et al., 12 Jan 2026), non-integer exponents dense in $|\cdot|$ 56 occur in the snowflake constructions (Bridson et al., 16 Dec 2025), and fibre-product methods yield examples with extremely fast primitive-recursive growth (Bridson et al., 29 Dec 2025). Yet a basic gap persists: the $|\cdot|$ 57-machine construction that realizes many functions always carries a quadratic error term, and the explicit open problem is whether there exists a finitely generated group $|\cdot|$ 58 such that

$|\cdot|$ 59

(Gillis et al., 12 Jan 2026). At the extreme end, the Baumslag–Gersten paper conjectures that no one-relator group has a larger conjugator length function than the Baumslag–Gersten group (Gillis, 29 Jul 2025).

Taken together, these results show that the conjugator length function has evolved from an effective refinement of conjugacy into a large organizing principle for geometric conjugacy theory. It records shortest-conjugator geometry, interacts in subtle ways with subgroup distortion and annular fillings, supports explicit algorithmic bounds, and exhibits a spectrum ranging from exact linear formulas to non-elementary growth (Sale, 2013).