Proper products and external returns (Appendix C.2) #
lem:nondeep-two-object-return and prop:proper-ray-products
(local-ray-restrictions.tex, primitive-ray-equations.tex).
The opposite form of prim_action_gen (Lemma 8.18): a nonzero target action makes the
Hom cyclic at the target.
The core of Lemma C.2 (lem:nondeep-two-object-return, first paragraph): for an external
return el = q_b^a with q_b^{a+1} ≠ 0, both Homs are generated only at b, and le = 0.
Lemma C.2(1) (lem:nondeep-two-object-return): an external return el = q_b^a with
q_b^{a+1} ≠ 0 has a = 2, N_b = 4 and le = 0; both Homs are generated only at b.
Left cancellation of a local power.
A left-sided generating ray leaves the return corner of its source.
Step 2 of the length bound: a reverse product f h = q_b^m of a left-sided generating f
comes from a generating reverse cyclic at b.
Positive comparisons in the faithful projective support at x.
Q_i ≤ Y_j fails for i > j: it would make the generating ray proper.
Proposition 8.25(1), length bound (Steps 1–3 of Appendix C.2): if a left-sided
generating ray f : b → y has q̂_b² f ≠ 0, then q̂_b³ = 0.
Proposition 8.25(1), length bound, right-sided form.
Endpoint layer preservation, generating inputs (Appendix C.2): for generating left-sided
f, h out of b, the product f v is never q̂_b h with q̂_b² h ≠ 0.
A nonidentity factor r₀ of a left-sided r = q̂_b^i r₀ is left-sided.
Proposition 8.25(1), endpoint layer preservation: if r = q̂_b^i r₀ with r₀
generating and rs are left-sided, then r₀ s is generating and rs = q̂_b^i (r₀ s).
A nondeep external return at b (def:nondeep-return).
Instances For
A generating two-sided ray with a nonzero reverse product reaches the return corner.
Proposition 8.25(4) (prop:proper-ray-products, Appendix C.2 "absence of external
two-sided action"): at the base of a nondeep external return there is no generating two-sided
ray to a vertex outside the return corner.
Proposition 8.25(2) (with N_x = 3: q̂_x² ≠ 0, q̂_x³ = 0): a product of generating rays of types B, L with proper
shiftable output q̂_x^k f has k = 1 and N_x = 3, and its first factor has a primitive
reverse.
Proposition 8.25(1), endpoint layer preservation, right-sided form.
A right-sided generating ray leaves the return corner of its target (when N ≥ 3).
Proposition 8.25(4), incoming form.
Proposition 8.25(3) (prop:proper-ray-products): a product of generating rays of
types L, R with proper shiftable output has endpoint layer two; it is a nondeep external
return if x = z, and otherwise factors through a nondeep return at one endpoint.
Deviation (Appendix C.2, Steps 1–4): with the retraction criterion of Lemma 8.9 the
complementary-path tree needs only the cover factorizations, which hold for j = 1
without further hypotheses; so j = 1 is excluded directly and the quotient A' of
Step 4 (with the induction hypothesis) is not used.
Proposition 8.25(2), return-corner form (prop:proper-ray-products(2)): in
BL_proper, the first factor r : x → y reaches the primitive return corner, y ∈ S_x.
Lemma C.2 (lem:nondeep-two-object-return): at a supporting
vertex b, generating rays e : b → y, l : y → b with y ∉ S_b, el = q̂_b^a and
q̂_b^{a+1} ≠ 0 have types L and R, N_b = 4, el = q̂_b² and le = 0.