NONDIVERGENCE OF REDUCTIVE GROUP ACTION ON HOMOGENEOUS SPACES

The paper’s Theorem 1.5 proves the equivalence (1)⇔(1′)⇔(2)⇔(3) for H = AM (with A ⊂ D ⊂ ZG(M) R-split, M semisimple, D ⊂ T after conjugacy), via (3)⇒(1) by a compactness/discreteness argument (Proposition 2.4), (1)⇒(2) by a topological covering theorem à la McMullen–Solan–Tamam, and (2)⇒(3) by a highest-weight/wedge-Ad construction that uses both Pi and τ(Pi) and Weyl groups W(G), W(ZG(M)) . The candidate solution establishes the same equivalence but follows a different route: it (i) builds a finite set of “height” functions from the fundamental-weight wedge-Ad representations and a finite cusp set (using reduction theory G = K S^t C F^{-1} Γ and the ∧di Ad models of fundamental weights) and asserts 1⇔3 directly via a Mahler–Borel criterion; (ii) proves 2⇒3 via a product-of-heights argument using linear dependence on Lie(A) and the fact that M acts trivially on the highest-weight line; and (iii) sketches 1⇒2 by passing to a single cusp/Weyl-face and arguing that failure of uniform nondivergence forces w−1Mw ⊂ ⋂Pi ∩ ⋂τ(Pi) and a dependence among restricted weights. These steps match the spirit and conclusions of the paper, though the candidate compresses (1)⇒(2) (the paper uses a nontrivial covering theorem) and appeals to a standard compactness criterion not explicitly stated in the paper. The equivalence (1)⇔(1′) is handled in the paper via Benoist–Quint’s random walk method; the candidate notes it informally but the logical reduction through (2) and (3) indeed makes 1′⇔1 in this setting . Overall, both are correct; the proofs are conceptually related (fundamental weights, wedge-Ad, cusps), but the paper’s argument relies on a topological covering theorem and a careful use of τ(Pi), whereas the candidate gives a streamlined height-function proof for 1⇔3 and a sketch for 1⇒2.