[Paper Review] A Height Gap Theorem For Finite Subsets Of GL_d(\bar{Q}) and Non Amenable Subgroups

Emmanuel Breuillard|arXiv (Cornell University)|Apr 9, 2008

Geometric and Algebraic Topology22 references3 citations

TL;DR

This paper introduces a conjugation-invariant normalized height function $\widehat{h}(F)$ for finite subsets $F$ of $GL_d(\overline{\mathbb{Q}})$, proving a uniform height gap: $\widehat{h}(F) > \varepsilon(d) > 0$ whenever $F$ generates a non-amenable subgroup. The result generalizes the Lehmer conjecture to matrix groups and provides a global adelic analog of Margulis' lemma, with applications to uniform Burnside-Schur theorems and the strong Tits alternative.

ABSTRACT

We show a global adelic analog of the classical Margulis Lemma from hyperbolic geometry. We introduce a conjugation invariant normalized height $\hat{h}(F)$ of a finite set of matrices $F$ in $GL_{n}(\bar{\Bbb{Q}})$ which is the adelic analog of the minimal displacement on a symmetric space. We then show, making use of theorems of Bilu and Zhang on the equidistribution of Galois orbits of small points, that $\hat{h}(F)>ε$ as soon as $F$ generates a non-virtually solvable subgroup of $SL_{n}(\bar{\Bbb{Q}}),$ where $ε=ε(n)>0$ is an absolute constant.

Motivation & Objective

• To define and study a conjugation-invariant normalized height $\widehat{h}(F)$ for finite subsets $F \subset GL_d(\overline{\mathbb{Q}})$, capturing the asymptotic spectral growth of $F^n$ across all places.
• To establish a uniform lower bound $\widehat{h}(F) > \varepsilon(d) > 0$ when $F$ generates a non-amenable subgroup, extending the Lehmer problem to linear groups.
• To provide a global adelic analog of Margulis' lemma by linking the height gap to the geometric and arithmetic behavior of matrix semigroups.
• To apply the height gap to prove a uniform version of the Burnside-Schur theorem on torsion linear groups.
• To lay the foundation for a strong uniform version of the Tits alternative in subsequent work.

Proposed method

• Define the normalized height $\widehat{h}(F) = \lim_{n \to \infty} \frac{1}{n} h(F^n)$, where $h(F)$ is a global arithmetic height summing local operator norms over all places of a number field.
• Use the spectral radius formula to relate $\widehat{h}(F)$ to the Weil height of eigenvalues when $F$ is a singleton, linking to classical Diophantine geometry.
• Apply local estimates on Chevalley groups over local fields to control the growth of $||F^n||_v$ at each place $v$, using the structure of $p$-adic and archimedean norms.
• Employ the Northcott property to show that bounded $\widehat{h}(F)$ and bounded degree imply finitely many conjugacy classes, under semisimplicity assumptions.
• Use the action on the Bruhat-Tits building and fixed-point properties in $p$-adic fields to analyze the displacement of $F$-orbits and derive lower bounds on $\widehat{h}(F)$.
• Apply the Cayley-Hamilton theorem and representation theory to reduce the problem to faithful representations over $\overline{\mathbb{Q}}$, especially in the case of virtually solvable or unipotent subgroups.

Experimental results

Research questions

• RQ1 Can a uniform lower bound on the normalized height $\widehat{h}(F)$ be established for finite subsets $F \subset GL_d(\overline{\mathbb{Q}})$ generating non-amenable subgroups?
• RQ2 How does the normalized height $\widehat{h}(F)$ relate to the spectral properties of $F$ and the Zariski closure of the group it generates?
• RQ3 To what extent does the height gap phenomenon generalize the classical Lehmer conjecture to non-abelian linear groups?
• RQ4 Can the height gap be used to derive uniform bounds in problems like the Burnside-Schur theorem on torsion linear groups?
• RQ5 What is the relationship between the normalized height $\widehat{h}(F)$ and the geometric action of $F$ on the Bruhat-Tits building over $p$-adic fields?

Key findings

• There exists an absolute constant $\varepsilon(d) > 0$ such that $\widehat{h}(F) > \varepsilon(d)$ whenever $F$ generates a non-amenable subgroup of $GL_d(\overline{\mathbb{Q}})$.
• The normalized height $\widehat{h}(F)$ vanishes if and only if $F$ generates a quasi-unipotent subgroup, i.e., all eigenvalues of elements in the group are roots of unity.
• The normalized height satisfies $\widehat{h}(F^n) = n \cdot \widehat{h}(F)$, and it is invariant under conjugation in $GL_d(\overline{\mathbb{Q}})$.
• A finite set $F$ can always be conjugated into a position where its standard height $h(F)$ is comparable to $\widehat{h}(F)$, with a constant depending only on $d$.
• The height gap implies a uniform version of the Burnside-Schur theorem: any finitely generated torsion subgroup of $GL_d(\overline{\mathbb{Q}})$ is finite, with a bound depending only on $d$.
• The result provides a global adelic analog of Margulis' lemma, where the height gap corresponds to a uniform lower bound on displacement in the Bruhat-Tits building.