Relatively Anosov Representations via Flows I: Theory

The paper proves the equivalence between (A) boundary–map relative Anosov, (B) existence of a contracting flow for some weak cusp space, and (C) a contracting flow for every Groves–Manning cusp space (Theorem 1.3) using a direct flow–bundle construction on G(X) and a careful thick–thin analysis of Groves–Manning cusps, together with singular value growth for peripheral subgroups and a dominated-splitting equivalence on Hom-bundles (Definition 1.2, Proposition 4.9, Theorem 9.1, Proposition 13.7) . The candidate solution outlines a different, standard approach: (A)⇒(C) via KLP’s relativized uniform gap along relative geodesics plus BCLS/BPS graph-transform arguments; (B)⇒(A) by deducing the strong dynamics preserving property from exponential contraction; and (C)⇒(B) trivially. This is consistent with known theory, albeit at a sketch level, and slightly overreaches by asserting independence across all weak cusp models via Mineyev’s flow space (not needed for the theorem and not shown in the paper). Net: both are correct; the paper’s proof is complete and self-contained for the stated equivalence; the model gives a plausible alternative sketch.