Lemma 41 ([Boundary maps from sublinear height control]). Let F belong to the fixed Lipschitz quasi-isometry family, let A=A_F , and let β be a selected target ray representative. Put α=β A . There is a homeomorphism f_β:L_α⟶Lβ and a …

Since A^* permutes the generator rays, the source quotient is defined. Choose v∈ d with α(πv)<0 . There are only finitely many ray and label choices, so choose all these v 's once. and a uniform compact shadow. For x_t=x_D(tv) , Proposition 40 ([Uniform sublinear height estimate]) gives β(πF(x_t)-πF(x)) =tα(πv)+o(t)≤-c_0t (t≥T_0), with c_0,T_0>0 uniform in F,x and the finite choices. Set y_t=F(x)^ -1 F(x_t) . Its distance from 1 is O(1+t) . By Lemma 11 ([Path packing]) it has a lift y_t=n_td_t with |d_t|_D≤C(1+t) . The increments y_t^ -1 y_ t+1 have uniformly bounded length and hence bounded lifts. Equations and imply \|(p_β(y_t)^ -1 p_β(y_ t+1 ))\| ≤C(1+t)^M e^ -c_1t . The series is summable. In a left invariant Riemannian metric, d(1, Z)≤C\|Z\| , by using the path s↦(sZ) . Thus the projected trajectory at integer times is Cauchy and stays in a fixed closed ball. Completeness and properness of the metric give convergence in a fixed compact set. The same estimate for time increments of size at most one gives convergence for real t→∞ . The initial interval 0≤t≤T_0 contributes a fixed compact set, so the limit lies in a fixed compact Q_β . Equivariance of p_β under the G -action therefore gives a limit b(x)=_ t→∞ p_βF(x_D(tv)) ∈F(x)·Q_β. of the representative. We first record the path estimate used for this purpose. Suppose, for large t , a source path stays at distance at most C_2t from a fixed x , has length bounded by a polynomial in t , and satisfies α(π(x^ -1 z))≤-c_2t at each of its points z . The uniform sublinear estimate puts its image, relative to F(x) , at β -heights at most -c_2t/2 for all sufficiently large t . The lifts at all image points have D -length O(t) , because their distance from F(x) is O(t) . Subdivide the source path into bounded-length pieces. Equations and bound each projected image increment by a polynomial times e^ -ct . There are only polynomially many pieces, so the projected distance between the endpoints tends to zero. Constants here may depend on this fixed collection of paths. For fixed d∈D , connect x(tv) to xd(tv) by xd(s)(tv) , where d(s) is a fixed piecewise smooth path in D from 1 to d . Its length is polynomial in t , since conjugation in the nilpotent group D has polynomial norm. Its points have distance O(t) from x , and their relative α -heights equal tα(πv)+O(1) . The preceding path estimate gives b(xd)=b(x) . For fixed u∈I_α , we will vary the height within the halfspace α<0 to realize u while keeping the α -height strongly negative. Choose a compact convex set E⊂W containing πv , entirely in α<0 , such that every bracket word containing an outside-ray letter has h_Z(E)>0 . Here is why such an E exists. Every generator form outside the positive α -ray is unbounded above on the halfspace α≤α(πv) . For each outside form choose a point in that halfspace on which it is as large as desired, and take the convex hull together with πv . The support maxima for inside-ray letters remain fixed negative numbers. Since the list of words and their lengths is finite, sufficiently large outside maxima make the support sum of every mixed word positive. By Lemma 35 ([Cone quotients and their Haar multipliers]), the mixed words span (I_α) . Thus the fixed element u belongs to B_t(E,O(1)) for all sufficiently large t . One can see this directly in the all-word description of the box: the fixed coordinates of u in a mixed-word basis are dominated by its positive exponential widths. Choose an affine section σ:W→V of π with σ(πv) equal to the abelianization of v . The compact convex set E_V=σ(E) contains that abelianization and satisfies πE_V=E . The paths of Lemma 10 ([Paths visiting support positions]), applied to E_V and the pivot (tv) , connect (tv) to u(tv) in length O(t) with heights in t(E+o(1)) . All their heights remain strictly negative in α by a linear amount, and all their points have distance O(t) from 1 . Left translation by x and the preceding path estimate give b(xu)=b(x) . The stabilizer of 1∈L_α is I_αD . The two invariances therefore show that b(x) depends only on p_αx . Define f_β(p_αx)=b(x) . This establishes. and continuity. By Proposition 40 ([Uniform sublinear height estimate]), A_ F =A^ -1 , so the construction for F gives f:L_β→L_α . For x_t=x(tv) the quotient point p_αx_t is fixed. Equation at x_t supplies representatives z_t of f_β(p_αx) within a fixed distance of F(x_t) : lift the compact set Q_β to a bounded set in G . The coarse-inverse estimate and the shadow estimate for f then give f(f_β(p_αx))∈x_t·Q' for a fixed compact Q'⊂L_α . The sets on the right contract to p_αx , since α(πv)<0 . Thus f f_β= . Interchanging the maps proves f_β f= . Fix x and t . The projection of a fixed-radius neighborhood of x_t contains a neighborhood of p_αx ; this uses only that the quotient projection is a submersion. By the Lipschitz bound and, its image under f_β is contained in F(x_t)·Q'' , with Q'' fixed. These sets have diameter tending to zero. Indeed, after normalization by F(x) , the D -part has length O(t) and negative linear β -height, so applies; quotient left translations are isometries. The fixed action of F(x) is Lipschitz, being a left translation followed by an automorphism. This proves continuity at p_αx . The same argument proves continuity of f , completing the proof.