The Structures of Higher Rank Lattice Actions on Dendrites

The uploaded paper states exactly the two structural conclusions for higher-rank lattice actions on dendrites with no infinite-order points (inverse-limit model with monotone bonding maps; factor r with the endpoint/infinite-orbit and shrinking-fiber properties) as Theorem 1.1, and indicates the proof reduces to a fixed-arc lemma (Theorem 4.3) and a finiteness-of-index-subgroups fact (Lemma 5.1), with the remainder carried out along the lines of the authors’ earlier work . The candidate solution reconstructs that same strategy in detail: it uses the pointwise fixed-arc lemma (matching Theorem 4.3 of the paper) to build an increasing sequence of Γ-invariant subdendrites with finite-quotient actions and monotone first-point bonding maps, identifies the inverse limit, and derives the endpoint/infinite-orbit and shrinking-fiber properties. The only substantive discrepancy is that the model asserts a Γ-fixed point outright; the paper only requires a finite orbit and then works with finite-index normal subgroups fixing points, as also enabled by Lemma 5.1 and used in the inductive construction. This can be repaired in the model’s writeup without changing the core argument. Hence both are correct, and the proofs are substantially the same, with the paper tersely deferring well-known inverse-limit details to earlier work while the model makes those steps explicit.