[Paper Review] A note on approximate subgroups of GL_n(C) and uniformly nonamenable groups

Emmanuel Breuillard, Ben Green+1 more|arXiv (Cornell University)|Jan 13, 2011

Finite Group Theory Research10 references10 citations

TL;DR

This paper provides a new, effective proof that any $K$-approximate subgroup of $\mathrm{GL}_n(\mathbb{C})$ is $\exp(K^{O(\log K)})$-controlled by a nilpotent $K^{O(1)}$-approximate group, using the uniform Tits alternative and additive combinatorics. The method extends to uniformly nonamenable groups, yielding similar control bounds with dependence on the uniform nonamenability parameter $\kappa$. The result is effective in principle, with explicit dependence on $K$ and $n$, though constants remain non-trivial to compute.

ABSTRACT

The aim of this brief note is to offer another proof of a theorem of Hrushovski that approximate subgroups of GL_n(C) are almost nilpotent. This approach generalizes to uniformly non amenable groups.

Motivation & Objective

• To provide an effective, non-model-theoretic proof of Hrushovski's theorem on approximate subgroups in $\mathrm{GL}_n(\mathbb{C})$.
• To establish explicit polynomial and exponential bounds on the control of approximate subgroups by nilpotent subgroups.
• To generalize the result to uniformly nonamenable groups, extending the control framework beyond linear groups.
• To demonstrate that the $O_{K,n}(1)$ term in control bounds can be made effective, with explicit dependence on $K$ and $n$, using additive combinatorics and uniform alternatives.
• To bridge the gap between qualitative model-theoretic results and quantitative, computable bounds in approximate subgroup theory.

Proposed method

• Apply the uniform Tits alternative (Proposition 2.1) to show that any finite symmetric set in $\mathrm{GL}_n(\mathbb{C})$ either generates a virtually solvable group or contains a free subgroup.
• Use Sanders’ and Croot–Sisask’s results (Proposition 2.2) to find a large subset $A'$ and a set $B$ such that $|A'B^m| \leq (1+\varepsilon)|A'|$ for small $\varepsilon$, implying $B^m$ cannot generate a free group.
• Apply Proposition 2.3 to deduce that if $|A'B^m| \leq (1+\varepsilon)|A'|$, then $B^m$ cannot contain two elements generating a nonabelian free group.
• Use the uniform Tits alternative to conclude that $B$ generates a virtually solvable group $G$, and apply Mal’cev–Platonov’s theorem (Proposition 2.4) to find a solvable subgroup $H \leq G$ of bounded index $[G:H] = O_n(1)$.
• Apply the pigeonhole principle to find a coset $Hx$ intersecting $B$ in a large set, so $|A^4 \cap Hx| \gg_{n,K} |A|$, enabling application of Ruzsa-type covering via Lemma 2.6.
• Use Lemma 2.6 to show that $A^2 \cap H$ is a $2K^3$-approximate solvable group that $O_{K,n}(1)$-controls $A$, and then apply Proposition 2.5 to upgrade this to a nilpotent $K^{O(1)}$-approximate group.

Experimental results

Research questions

• RQ1 Can the control of approximate subgroups in $\mathrm{GL}_n(\mathbb{C})$ by nilpotent subgroups be proven without model theory, with explicit bounds?
• RQ2 What is the quantitative dependence of the control term $O_{K,n}(1)$ on $K$ and $n$ in the context of approximate subgroups?
• RQ3 How does the uniform Tits alternative enable effective bounds in the classification of approximate subgroups?
• RQ4 Can the framework of control by amenable approximate groups be extended to uniformly nonamenable groups?
• RQ5 What is the optimal bound for control in hyperbolic groups, and how does it compare to the general bound derived here?

Key findings

• The paper establishes that any $K$-approximate subgroup $A \subseteq \mathrm{GL}_n(\mathbb{C})$ is $\exp(K^{O(\log K)})$-controlled by a nilpotent $K^{O(1)}$-approximate group, with the exponent depending on the uniform Tits alternative constant $m(n)$.
• The bound $O_{K,n}(1)$ in the control term is effective in principle, with explicit dependence on $K$ and $n$, derived from Sanders’ and Croot–Sisask’s results.
• The proof avoids ultrafilters and model theory, offering a quantitative alternative to Hrushovski’s original result.
• The method generalizes to $\kappa$-uniformly nonamenable groups, showing that $A$ is $\exp(K^{O(\log K / \kappa)})$-controlled by an amenable $K^{O(1)}$-approximate group.
• For $\delta$-hyperbolic groups, the result implies control by cyclic subgroups, though the bound is likely improvable to $O_\varepsilon(K^{2+\varepsilon})$ as in Safin’s work.
• The proof relies on a combination of additive combinatorics (Ruzsa covering, product set estimates) and structural group theory (virtually solvable subgroups, Mal’cev–Platonov theorem).