---
title: Rank-Corrected Inverse-Squared Gap Function
url: https://www.emergentmind.com/topics/rank-corrected-inverse-squared-gap-function
type: topic
---

# Rank-Corrected Inverse-Squared Gap Function

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" [1212.6749], 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 $\beta_r$.

## 1. Definitions and invariant formulation

Let $G$ be any finitely generated group with a fixed finite generating set $A=\{a_1,\dots,a_r\}$. If $|g|_A$ denotes the word-length of $g\in G$ with respect to $A$, then the size of an automorphism $\phi\in\operatorname{Aut} G$ is defined by
$$
\|\phi\|_A=\sum_{i=1}^r |a_i\phi|_A.
$$
For an outer automorphism $[\phi]\in\operatorname{Out} G$, the corresponding size is
$$
\|[\phi]\|_A=\min\{\|\psi\|_A:\psi\in\operatorname{Aut} G,\ [\psi]=[\phi]\}.
$$
From these norms one defines the gap functions
$$
\alpha_A(n)=\max\{\|\phi^{-1}\|_A:\phi\in\operatorname{Aut} G,\ \|\phi\|_A\le n\},
$$
and
$$
\beta_A(n)=\max\{\|[\phi]^{-1}\|_A:[\phi]\in\operatorname{Out} G,\ \|[\phi]\|_A\le n\},
$$
with the convention that the maximum of the empty set is $0$, so $\alpha_A(n)=\beta_A(n)=0$ for $n$ below the rank [1212.6749].

A standard argument shows that the equivalence classes of $\alpha_A$ and $\beta_A$ do not depend on the choice of $A$, where
$$
f\lesssim g \iff \exists C>0:\ f(n)\le C\cdot g(Cn)\ \text{and}\ g(n)\le C\cdot f(Cn).
$$
The resulting group-invariants are denoted $\alpha_G(n)$ and $\beta_G(n)$. For the free group $F_r$ with chosen free basis $A_r$, the specialized notation is
$$
\alpha_r(n)=\alpha_{F_r}(n),\qquad \beta_r(n)=\beta_{F_r}(n).
$$
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 [1212.6749] gives:
- for all $n\ge4$,
  $$
  \alpha_2(n)<(n-1)^2;
  $$
- for all $n\ge10$,
  $$
  \alpha_2(n)>n^2-6n+42;
  $$
- hence
  $$
  \alpha_2(n)\asymp n^2;
  $$
- for all $n\ge0$,
  $$
  \beta_2(n)=n,
  $$
  and therefore
  $$
  \beta_2(n)\asymp n.
  $$

These statements isolate two distinct phenomena. First, for $\operatorname{Aut} F_2$, the worst-case inverse complexity is exactly quadratic. Second, for $\operatorname{Out} F_2$, the corresponding behavior is exactly linear. In the terminology used in the source, $F_2$ therefore exhibits a precise inverse-squared gap for $\operatorname{Aut} F_2$, while the outer-automorphism analogue is strictly smaller [1212.6749].

The rank-two case is also the only regime in which the lower and upper asymptotic estimates meet exactly. This makes rank $2$ 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 $r\ge3$, the exact growth of $\alpha_r(n)$ and $\beta_r(n)$ is not pinned down. The higher-rank theorem establishes the existence of constants $K_r,K_r',M_r>0$ such that for all $n\ge0$,
$$
K_r\cdot n^r\le \alpha_r(n),
$$
and
$$
K_r'\cdot n^{r-1}\le \beta_r(n)\le M_r\cdot n^{M_r}.
$$
Accordingly, $\beta_r$ grows at least like $n^{r-1}$ but no faster than some fixed polynomial $n^{M_r}$, while $\alpha_r$ grows at least like $n^r$ [1212.6749].

This is the sense in which the inverse-squared phenomenon becomes rank-corrected. The exponent $2$ visible in $\alpha_2$ does not persist uniformly across all ranks. Instead, the lower-bound exponent rises with rank: at least $r$ for automorphisms and at least $r-1$ for outer automorphisms. The data support the symbolic summary
$$
\alpha_r(n)\gtrsim n^r,\qquad \beta_r(n)\gtrsim n^{r-1},\qquad \beta_r(n)\lesssim n^{M_r},
$$
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 $2$.

## 4. Lower bounds via abelianization

The lower bounds are obtained from an explicit one-parameter family of positive automorphisms
$$
\Phi_p\in \operatorname{Aut}^+ F_r,\qquad p\ge2,
$$
whose abelianization is a unipotent matrix $M(p)\in GL_r(\mathbf Z)$ with diagonal entries $1$ and superdiagonal entries $p$. A direct calculation yields
$$
\|\Phi_p\|_1=r+(r-1)\cdot p\lesssim p,
$$
and
$$
\|\Phi_p^{-1}\|_1\gtrsim p^{r-1}.
$$
After choosing $p\asymp n$, this gives
$$
\beta_r(n)\gtrsim n^{r-1}.
$$
To raise the lower bound for $\alpha_r(n)$ to $n^r$, one conjugates $\Phi_p$ by suitable elements in $F_r$ so as to bootstrap an extra power of $p$ [1212.6749].

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 $\beta_r$ to $\alpha_r$ 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 $\beta_r$ is obtained through Culler-Vogtmann Outer space $X_r$ equipped with the asymmetric Lipschitz metric $d(-,-)$. A theorem of Algom-Kfir and Bestvina states that on the $\varepsilon$-thick part of $X_r$ the metric is quasi-symmetric:
$$
d(x,y)\lesssim d(y,x),
$$
up to a constant depending only on $r$ and $\varepsilon$. Specializing to the marked rose of volume $1/r$, one identifies
$$
d(x,\phi\cdot x)=\log\|\phi\|,\qquad d(\phi\cdot x,x)=\log\|\phi^{-1}\|.
$$
This yields the inequality
$$
\log\|\phi^{-1}\|\lesssim \log\|\phi\|.
$$
Passing back to $\beta_r(n)$ and absorbing constants gives the polynomial upper bound
$$
\beta_r(n)\lesssim n^{M_r}
$$
for some $M_r$ [1212.6749].

This argument is specific to the outer-automorphism setting. The data do not provide a corresponding polynomial upper bound for $\alpha_r$, 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 | $\alpha_r(n)$ | $\beta_r(n)$ |
|---|---|---|
| $r=2$ | $\asymp n^2$ | $=n$ |
| $r\ge3$ | $\gtrsim n^r$ | $\gtrsim n^{r-1}$ and $\lesssim n^{M_r}$ |

Two clarifications are central. First, although the norms $\|\phi\|_A$ and $\|[\phi]\|_A$ depend on the chosen generating set, the equivalence classes of the induced gap functions do not; $\alpha_G$ and $\beta_G$ 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 $\alpha_2$, while the higher-rank formulation replaces exponent $2$ by exponents controlled by rank itself [1212.6749].

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 $2$, $\alpha_2$ is quadratic whereas $\beta_2$ is linear. In higher rank, the lower bounds likewise differ by one power of $n$. The terminology “rank-corrected inverse-squared gap function” refers precisely to this family of estimates: exact exponent $2$ in rank $2$ for $\alpha_2$, exact exponent $1$ in rank $2$ for $\beta_2$, and higher-rank lower bounds corrected to exponents $r$ and $r-1$ respectively.

Source: https://www.emergentmind.com/topics/rank-corrected-inverse-squared-gap-function