Mixed identities in linear groups -- effective version

TL;DR

We study mixed identities in linear groups through the notion of MIF (mixed-identity-free) groups and establish that linear MIF groups are sharply MIF and linearly MIF. The approach combines heights, random homomorphisms, generating pairs, and random walks with a probabilistic variant of (strong/super) approximation to obtain effective bounds: for a finite generating set $X$, the witnesses satisfy $f_X(n) \le C \log n$ and $\phi_X(n) \le C n$, and sharp MIF implies linearly MIF. A self-contained proof of strong approximation is given, alongside a probabilistic SSA result that works with arbitrary matrix entries, enabling exponential decay bounds for nontrivial mixed words along random walks. Consequences include trivial amenable radical for MIF linear groups, existence of a faithful representation with center-free semisimple Zariski closure, and primitivity; the work also connects to operator-algebraic regularity via reduced $C^*$-algebras. Overall, the paper advances understanding of identities in linear groups and provides a robust, effective probabilistic framework with potential applications to expansion theory and noncommutative geometry.

Abstract

We show that MIF (mixed-identity-free) linear groups are sharply MIF and linearly MIF. Along the way we provide a self contained proof of the strong approximation theorem, and a new (probabilistic) variant of the super approximation theorem.

Theorems & Definitions (32)

• Theorem 1.1
• Proposition 1.2
• Conjecture 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3