[Paper Review] Cheeger constant and algebraic entropy of linear groups

Emmanuel Breuillard, Tsachik Gelander|ArXiv.org|Jul 21, 2005

Advanced Operator Algebra Research4 references4 citations

TL;DR

This paper establishes a uniform version of the Tits alternative for linear groups, proving that non-virtually solvable subgroups of $\mathrm{GL}_n(K)$ contain free subgroups generated within a uniformly bounded word length. Using geometric and arithmetic tools in symmetric spaces and Bruhat-Tits buildings, the authors derive uniform lower bounds for the Cheeger constant and algebraic entropy of Cayley graphs, extending results to arbitrary characteristic and resolving uniformity in growth and spectral gaps.

ABSTRACT

We prove a uniform version of the Tits alternative. As a consequence, we obtain uniform lower bounds for the Cheeger constant of Cayley grahs of finitely generated non virtually solvable linear groups in arbitrary characteristic. Also we show that the algebraic entropy of discrete subgroups of a given Lie group is uniformly bounded away from zero.

Motivation & Objective

• To establish a uniform version of the Tits alternative for finitely generated linear groups over arbitrary fields.
• To derive uniform lower bounds for the Cheeger constant of Cayley graphs of non-virtually solvable linear groups.
• To show that the algebraic entropy of discrete subgroups of a given Lie group is uniformly bounded away from zero.
• To extend the Eskin-Mozes-Oh result on uniform exponential growth to include uniform spectral and geometric invariants.
• To unify geometric and arithmetic methods in the study of growth and spectral gaps in linear groups.

Proposed method

• Use the Borel–Harish-Chandra theorem to reduce the general case to arithmetic lattices in semisimple Lie groups.
• Apply the theory of symmetric spaces and Bruhat-Tits buildings to analyze the dynamics of group elements.
• Employ $KAK$-decompositions and Lipschitz estimates to characterize contracting and very proximal elements.
• Use the pigeonhole principle and arithmetic complexity bounds to find conjugating elements that separate attracting and repelling directions.
• Leverage Mostow and Landvogt’s theorems on totally geodesic embeddings in CAT(0) spaces to lift elements from the Lie group to the arithmetic lattice.
• Establish uniform bounds via the interplay between geometric separation and arithmetic complexity of algebraic vectors.

Experimental results

Research questions

• RQ1 Can the Tits alternative be strengthened to a uniform bound on the word length required to find independent generators in non-virtually solvable linear groups?
• RQ2 What is the uniform lower bound for the Cheeger constant of Cayley graphs of linear groups over arbitrary fields?
• RQ3 How does the algebraic entropy of discrete subgroups of a fixed Lie group behave uniformly across generating sets?
• RQ4 To what extent can spectral gaps (via Kazhdan constants) be uniformly bounded away from zero in linear groups?
• RQ5 Can geometric methods in symmetric spaces and buildings be used to prove uniform growth and spectral properties in arithmetic lattices?

Key findings

• The independence diameter $d_\Gamma$ is uniformly bounded for all finitely generated non-virtually solvable linear groups over any field.
• A uniform lower bound on the Cheeger constant $h_\Gamma$ is established for all Cayley graphs of such groups, independent of the generating set.
• The algebraic entropy of discrete subgroups of a given Lie group is uniformly bounded away from zero, implying uniform exponential growth.
• For non-uniform arithmetic lattices, the uniform bound on the independence diameter depends only on the ambient Lie group $G$, not on the lattice.
• The proof yields a uniform lower bound on the $\ell^2$-Kazhdan constant $\kappa_\Gamma$, ensuring uniform non-amenability.
• The construction of ping-pong pairs relies on arithmetic complexity and geometric separation, with bounds depending on the degree $d$ and dimension $n$.