[Paper Review] Generic thinness in finitely generated subgroups of $ extrm{SL}_n(\mathbb Z)$

Elena Fuchs, Igor Rivin|arXiv (Cornell University)|Jun 4, 2015

Geometric and Algebraic Topology15 references3 citations

TL;DR

This paper demonstrates that in the symmetrized Euclidean ball model of $\mathrm{SL}_n(\mathbb{Z})$, two randomly chosen elements generate a thin free group with probability tending to 1 as the radius $X \to \infty$, by showing they form a ping-pong pair on a suitable space. The result establishes generic thinness in this model, resolving a key question in the study of random subgroups of arithmetic groups.

ABSTRACT

We show that for any $n\geq 2$, two elements selected uniformly at random from a \emph{symmetrized} Euclidean ball of radius $X$ in $ extrm{SL}_n(\mathbb Z)$ will generate a thin free group with probability tending to $1$ as $X ightarrow \infty.$ This is done by showing that the two elements will form a ping-pong pair, when acting on a suitable space, with probability tending to $1$. On the other hand, we give an upper bound less than $1$ for the probability that two such elements will form a ping-pong pair in the usual Euclidean ball model in the case where $n>2$.

Motivation & Objective

• To determine whether a generic finitely generated subgroup of $\mathrm{SL}_n(\mathbb{Z})$ is thin under a natural Euclidean model of randomness.
• To resolve the discrepancy between combinatorial and Euclidean models of genericity in the context of thin groups.
• To establish that symmetrized random elements in $\mathrm{SL}_n(\mathbb{Z})$ form a ping-pong pair with high probability, enabling the proof of generic thinness.
• To extend the understanding of random subgroups in arithmetic groups beyond the combinatorial height model.

Proposed method

• Use a symmetrized Euclidean ball model $B_X'(G) = \{g \in \mathrm{SL}_n(\mathbb{Z}) \mid \|g\|, \|g^{-1}\| \leq X\}$ to define a probability measure $\mu_X'$ on pairs of generators.
• Apply Breuillard-Gelander's characterization of ping-pong pairs in $\mathrm{SL}_n(\mathbb{R})$ acting on projective space to analyze the dynamics of random matrices.
• Leverage singular value decomposition and concentration of measure to show that random matrices have large singular value gaps and uniformly distributed leading singular vectors with high probability.
• Use Lyapunov exponent estimates via random matrix products to show exponential divergence of orbits, supporting the ping-pong mechanism.
• Combine equidistribution of singular vectors and gap estimates to prove that the probability of forming a ping-pong pair tends to 1 as $X \to \infty$.
• Use the ping-pong argument to conclude that the generated group is free and thin in $\mathrm{SL}_n(\mathbb{Z})$.

Experimental results

Research questions

• RQ1 Does a generic 2-generator subgroup of $\mathrm{SL}_n(\mathbb{Z})$ generated by elements chosen uniformly from a symmetrized Euclidean ball of radius $X$ become thin as $X \to \infty$?
• RQ2 Can the ping-pong mechanism be used to prove generic thinness in the Euclidean model, despite its failure in the standard Euclidean ball model for $n > 2$?
• RQ3 How does the symmetrized ball model affect the probability of generating a free group via ping-pong dynamics?
• RQ4 What role do singular value gaps and vector alignment play in ensuring the ping-pong condition for random matrices in $\mathrm{SL}_n(\mathbb{Z})$?

Key findings

• For $n \geq 2$, two elements chosen uniformly at random from the symmetrized Euclidean ball $B_X'(G)$ form a ping-pong pair with probability tending to 1 as $X \to \infty$.
• The probability that two such elements form a ping-pong pair in the standard Euclidean ball model is strictly less than 1 when $n > 2$, indicating the necessity of symmetrization.
• The Lyapunov exponent of the group generated by two random elements in the symmetrized model tends to infinity with $X$, with high probability.
• The singular values of random matrices in $B_X'(G)$ exhibit a growing gap between the first and second singular values as $X \to \infty$.
• The leading singular vectors of random matrices in $B_X'(G)$ are uniformly distributed in the orthogonal group, with inner products bounded away from zero with high probability.
• As a consequence, the generated group is free and of infinite index in $\mathrm{SL}_n(\mathbb{Z})$, hence thin, with probability tending to 1.