Rigidity for Higher Rank Lattice Actions on Dendrites

The paper proves Theorem 1.1 (no almost-free action on a nondegenerate dendrite without infinite-order points) and Theorem 1.2 (for Γ ≤ SL_n(Z), n ≥ 3, existence of an invariant nondegenerate subdendrite with an almost-finite subsystem, and almost faithfulness if not finite). Its proofs pivot on: (i) Duchesne–Monod’s elementarity (fixed point or invariant arc) , (ii) Deroin–Hurtado’s non-left-orderability of higher-rank lattices , and (iii) a new characterization linking left-orderability to almost-free dendrite actions with a fixed endpoint (Proposition 4.1) . The explicit construction in Theorem 1.2 uses unipotent elements to produce arcs fixed pointwise (Proposition 5.3) and an inductive inverse-limit scheme to obtain an almost-finite action; Margulis’ normal subgroup theorem yields almost faithfulness when the action is not finite . The model’s solution follows the same strategy: it handles the invariant-arc and fixed-point cases as in the paper for (a), appeals to the same left-orderability characterization, and for (b) reconstructs the same unipotent-based arc-fixing step and the same inductive-inverse-limit argument. Minor presentational differences aside, the proofs are effectively the same as the paper’s.