Counting and boundary limit theorems for representations of Gromov-hyperbolic groups

The paper proves exponential large deviations for log‖ρ(g)‖ under uniform counting on spheres in non‑elementary hyperbolic groups (Theorem 1.6), via a boundary approach: first establish an almost-sure large deviation theorem for Patterson–Sullivan typical geodesic rays (Theorem 1.10), then transfer to counting; it also develops the Markov coding and Parry-like measures in detail and handles periodicity and multiple maximal components (Sections 2, 3, 6, 8) . The candidate’s solution proves the same counting large deviations by directly comparing uniform counting on length‑n geodesics with the Parry (maximal entropy) Markov law on the Cannon/shortlex automaton, and then invoking Bougerol’s large deviations for Markovian matrix products; this is a method explicitly noted as viable in Remark 6.3 (though not pursued there) . The model’s sketch omits technicalities that the paper handles carefully (non‑irreducibility and periodicity of the automaton; uniform Doeblin/minorization; handling operator vs vector norms), but these are standard fixable details (e.g., passing to a p‑step skeleton and decomposing across maximal components) and do not alter correctness. Hence both are correct, but the proofs are different in emphasis.