Limit spaces of vertex and edge replacement systems

The paper proves that if a VERS is n-expanding then its history graph H is Gromov hyperbolic by: (i) showing H is an augmented tree and defining canonical spanning lifts (Proposition 3.6 and Definition 3.7, , ); (ii) invoking Kaimanovich’s “no big squares” criterion for augmented trees (Theorem 3.10, ); (iii) proving a shrinking lemma: existence of a geodesic (n+1)-square implies a geodesic n-square (Lemma 3.11, ); and (iv) showing that n-expanding forbids geodesic n-squares (Theorem 3.12, ). The candidate solution mirrors these exact steps: it identifies H as an augmented tree with unique spanning lifts, uses the same Kaimanovich criterion, proves the same shrinking lemma via the grid/1-Lipschitz lifting argument, and derives the same contradiction to n-expanding by lifting a horizontal geodesic from a putative n-square. Apart from minor phrasing (top vs bottom side of the square and distinctness of lifts vs vertices), the logic and structure match the paper’s proof.