Rank-Corrected Inverse-Squared Gap Function

• The rank-corrected inverse-squared gap function quantifies the worst-case discrepancy between an automorphism (or outer automorphism) and its inverse using the complexity functions αₙ and βₙ.
• In rank 2, the function shows precise behavior with automorphisms growing quadratically (α₂ ~ n²) and outer automorphisms linearly (β₂ ~ n), setting a clear reference point.
• For ranks r ≥ 3, lower bounds indicate that αᵣ(n) grows at least as nʳ and βᵣ(n) as n^(r–1), reflecting increased inversion asymmetry and rank-sensitive behavior.

The rank-corrected inverse-squared gap function is the collection of asymptotic estimates that describes the worst-case discrepancy between the size of an automorphism or outer automorphism of a free group and the size of its inverse. In the framework of "Bounding the gap between a free group (outer) automorphism and its inverse" (Ladra et al., 2012), this discrepancy is encoded by two complexity functions, $\alpha_r$ for $\operatorname{Aut} F_r$ and $\beta_r$ for $\operatorname{Out} F_r$. The resulting picture is sharply rank-sensitive: in rank $2$, $\alpha_2(n)\asymp n^2$ and $\beta_2(n)\asymp n$, while for ranks $r\geqslant 3$ one has lower bounds $\alpha_r(n)\gtrsim n^r$ and $\beta_r(n)\gtrsim n^{r-1}$, together with a polynomial upper bound for $\operatorname{Aut} F_r$0.

1. Definitions and invariant formulation

Let $\operatorname{Aut} F_r$1 be any finitely generated group with a fixed finite generating set $\operatorname{Aut} F_r$2. If $\operatorname{Aut} F_r$3 denotes the word-length of $\operatorname{Aut} F_r$4 with respect to $\operatorname{Aut} F_r$5, then the size of an automorphism $\operatorname{Aut} F_r$6 is defined by

$\operatorname{Aut} F_r$7

For an outer automorphism $\operatorname{Aut} F_r$8, the corresponding size is

$\operatorname{Aut} F_r$9

From these norms one defines the gap functions

$\beta_r$0

and

$\beta_r$1

with the convention that the maximum of the empty set is $\beta_r$2, so $\beta_r$3 for $\beta_r$4 below the rank (Ladra et al., 2012).

A standard argument shows that the equivalence classes of $\beta_r$5 and $\beta_r$6 do not depend on the choice of $\beta_r$7, where

$\beta_r$8

The resulting group-invariants are denoted $\beta_r$9 and $\operatorname{Out} F_r$0. For the free group $\operatorname{Out} F_r$1 with chosen free basis $\operatorname{Out} F_r$2, the specialized notation is

$\operatorname{Out} F_r$3

Thus the rank-corrected inverse-squared gap is not an additional independently defined function; it is the collective asymptotic behavior of these two invariants in free-group rank.

2. Exact behavior in rank two

The rank-two case is completely determined up to the equivalence relation above. The theorem stated in (Ladra et al., 2012) gives:

• for all $\operatorname{Out} F_r$4,

$\operatorname{Out} F_r$5

• for all $\operatorname{Out} F_r$6,

$\operatorname{Out} F_r$7

• hence

$\operatorname{Out} F_r$8

• for all $\operatorname{Out} F_r$9,

$2$0

and therefore

$2$1

These statements isolate two distinct phenomena. First, for $2$2, the worst-case inverse complexity is exactly quadratic. Second, for $2$3, the corresponding behavior is exactly linear. In the terminology used in the source, $2$4 therefore exhibits a precise inverse-squared gap for $2$5, while the outer-automorphism analogue is strictly smaller (Ladra et al., 2012).

The rank-two case is also the only regime in which the lower and upper asymptotic estimates meet exactly. This makes rank $2$6 the reference point from which the later rank-corrected formulation is derived.

3. Higher-rank estimates and the rank correction

For free groups of rank $2$7, the exact growth of $2$8 and $2$9 is not pinned down. The higher-rank theorem establishes the existence of constants $\alpha_2(n)\asymp n^2$0 such that for all $\alpha_2(n)\asymp n^2$1,

$\alpha_2(n)\asymp n^2$2

and

$\alpha_2(n)\asymp n^2$3

Accordingly, $\alpha_2(n)\asymp n^2$4 grows at least like $\alpha_2(n)\asymp n^2$5 but no faster than some fixed polynomial $\alpha_2(n)\asymp n^2$6, while $\alpha_2(n)\asymp n^2$7 grows at least like $\alpha_2(n)\asymp n^2$8 (Ladra et al., 2012).

This is the sense in which the inverse-squared phenomenon becomes rank-corrected. The exponent $\alpha_2(n)\asymp n^2$9 visible in $\beta_2(n)\asymp n$0 does not persist uniformly across all ranks. Instead, the lower-bound exponent rises with rank: at least $\beta_2(n)\asymp n$1 for automorphisms and at least $\beta_2(n)\asymp n$2 for outer automorphisms. The data support the symbolic summary

$\beta_2(n)\asymp n$3

with the rank-two case providing the exact meeting point of the corresponding bounds.

A plausible implication is that the asymmetry between an automorphism and its inverse becomes more severe as rank increases, although the exact asymptotic exponent remains unresolved outside rank $\beta_2(n)\asymp n$4.

4. Lower bounds via abelianization

The lower bounds are obtained from an explicit one-parameter family of positive automorphisms

$\beta_2(n)\asymp n$5

whose abelianization is a unipotent matrix $\beta_2(n)\asymp n$6 with diagonal entries $\beta_2(n)\asymp n$7 and superdiagonal entries $\beta_2(n)\asymp n$8. A direct calculation yields

$\beta_2(n)\asymp n$9

and

$r\geqslant 3$0

After choosing $r\geqslant 3$1, this gives

$r\geqslant 3$2

To raise the lower bound for $r\geqslant 3$3 to $r\geqslant 3$4, one conjugates $r\geqslant 3$5 by suitable elements in $r\geqslant 3$6 so as to bootstrap an extra power of $r\geqslant 3$7 (Ladra et al., 2012).

This method identifies abelianization as the first source of large inverse norm. The additional conjugation step is what separates the automorphism bound from the outer-automorphism bound. In that sense, the passage from $r\geqslant 3$8 to $r\geqslant 3$9 records not merely matrix growth after abelianization, but also the extra distortion available before quotienting by inner automorphisms.

5. Polynomial upper bound for outer automorphisms

The upper bound for $\alpha_r(n)\gtrsim n^r$0 is obtained through Culler-Vogtmann Outer space $\alpha_r(n)\gtrsim n^r$1 equipped with the asymmetric Lipschitz metric $\alpha_r(n)\gtrsim n^r$2. A theorem of Algom-Kfir and Bestvina states that on the $\alpha_r(n)\gtrsim n^r$3-thick part of $\alpha_r(n)\gtrsim n^r$4 the metric is quasi-symmetric:

$\alpha_r(n)\gtrsim n^r$5

up to a constant depending only on $\alpha_r(n)\gtrsim n^r$6 and $\alpha_r(n)\gtrsim n^r$7. Specializing to the marked rose of volume $\alpha_r(n)\gtrsim n^r$8, one identifies

$\alpha_r(n)\gtrsim n^r$9

This yields the inequality

$\beta_r(n)\gtrsim n^{r-1}$0

Passing back to $\beta_r(n)\gtrsim n^{r-1}$1 and absorbing constants gives the polynomial upper bound

$\beta_r(n)\gtrsim n^{r-1}$2

for some $\beta_r(n)\gtrsim n^{r-1}$3 (Ladra et al., 2012).

This argument is specific to the outer-automorphism setting. The data do not provide a corresponding polynomial upper bound for $\beta_r(n)\gtrsim n^{r-1}$4, and that asymmetry is part of the present state of the theory described in the source.

6. Synthesis, scope, and recurrent points of confusion

The estimates can be summarized as follows.

Rank regime	$\beta_r(n)\gtrsim n^{r-1}$5	$\beta_r(n)\gtrsim n^{r-1}$6
$\beta_r(n)\gtrsim n^{r-1}$7	$\beta_r(n)\gtrsim n^{r-1}$8	$\beta_r(n)\gtrsim n^{r-1}$9
$\operatorname{Aut} F_r$00	$\operatorname{Aut} F_r$01	$\operatorname{Aut} F_r$02 and $\operatorname{Aut} F_r$03

Two clarifications are central. First, although the norms $\operatorname{Aut} F_r$04 and $\operatorname{Aut} F_r$05 depend on the chosen generating set, the equivalence classes of the induced gap functions do not; $\operatorname{Aut} F_r$06 and $\operatorname{Aut} F_r$07 are therefore group-invariants. Second, the rank-corrected inverse-squared gap should not be read as a claim that all ranks exhibit a literal square-law. The square-law is exact only for $\operatorname{Aut} F_r$08, while the higher-rank formulation replaces exponent $\operatorname{Aut} F_r$09 by exponents controlled by rank itself (Ladra et al., 2012).

A related source of confusion is the expectation that automorphisms and outer automorphisms should display the same inverse-growth behavior. The results show otherwise. Already in rank $\operatorname{Aut} F_r$10, $\operatorname{Aut} F_r$11 is quadratic whereas $\operatorname{Aut} F_r$12 is linear. In higher rank, the lower bounds likewise differ by one power of $\operatorname{Aut} F_r$13. The terminology “rank-corrected inverse-squared gap function” refers precisely to this family of estimates: exact exponent $\operatorname{Aut} F_r$14 in rank $\operatorname{Aut} F_r$15 for $\operatorname{Aut} F_r$16, exact exponent $\operatorname{Aut} F_r$17 in rank $\operatorname{Aut} F_r$18 for $\operatorname{Aut} F_r$19, and higher-rank lower bounds corrected to exponents $\operatorname{Aut} F_r$20 and $\operatorname{Aut} F_r$21 respectively.