Lemma 80 ([Relative row geometry]). Under the local hypotheses just listed, each folded vertex row Y_v is hyperbolic relative to its finite induced peripheral system, and each adjacent edge map Y_f↪Yv is an exact relative embedding. The …

Each Y_v is hyperbolic relative first to just its included S_d 's: glue coned fine hyperbolic graphs for its actual X -nodes along the single apex points corresponding to their S_d -neighbors. In X these use the nontrivial malnormal quasiconvex incidence groups. This is a tree of graphs glued at points by Γ_v ; hyperbolicity and fineness follow piecewise, with finite edge orbit set and finite edge stabilizers. The infinite vertex stabilizers of this glued graph are exactly the included object groups, in their actual coset positions; ordinary vertices have finite stabilizers. Thus gives a finite relative presentation with linear relative Dehn function. This comparison requires neither absolute finite generation of Y_v nor finite generation of an S_d . The same comparison applies to the edge-row model below with its exact induced apex stabilizers. A row equal to a single peripheral has the degenerate relative presentation with no ordinary generators or relators; a trivial row is immediate. The edge group Y_f is relatively hyperbolic and cone-quasi-isometrically embedded with its induced peripheral structure for this list. Indeed use K_f , gluing the corresponding relative models of F_x (with all nontrivial intersections with incident M 's) at the included points. A nontrivial S_d -intersection for d∈Γ_v-K_f is witnessed adjacent to K_f , in the corresponding actual piece (project from d to K_f , using strictness). The models at X embed relatively quasi-isometrically by ordinary quasiconvexity of F_x . One proof uses coarse nearest projection onto this quasiconvex orbit in X : projections of an incident M -coset have bounded diameter if the coset stays far away, otherwise lie boundedly near an orbit of the exact intersection. This follows by thin quadrilaterals and quasiconvexity, then ordinary properness (normalize a close pair, and pairs at bounded distance have only finitely many orbits under the intersection). Thus projections coarsely bound every cone jump in the induced cone metric. The intersection system itself is malnormal and quasiconvex there, finite up to the subgroup. Summing distances over the point gluings preserves the relative embedding (each internal passage between distinct gluing apices costs positively). These models are equivalent to relative Cayley metrics by relative cocompactness or normal paths along the Bass tree. External cyclic peripheral intersections to be added in Y_v are maximal loxodromic cyclics there, disjointly malnormal. Indeed root-closedness and disjointness are inherited (including for the extra family), these groups meet no included conjugate nontrivially, and loxodromic elementary closures are virtually cyclic, hence cyclic here. Thus adding them still gives relative hyperbolicity by elementary-subgroup peripheral enlargement~. For finitely many additions, iterate this statement over the distinct maximal cyclic types: a remaining type cannot become conjugate into an earlier one without being commensurable with it. The relative embedding above persists with the induced intersections, as we now prove separately. Use coned Cayley graphs (ordinary T -edges for finite relative generators, radial edges to coset apices). An embedded subgroup's cone path is mapped with fixed representative offsets p_α for its peripheral list B∩p_αP_i p_α^ -1 , via b,bp_α,bp_αP_i at a jump. We can enlarge T by all offsets and ordinary letters needed, and also by a generator for each new loxodromic cyclic subgroup, whose translates now have uniform ordinary-edged quasigeodesic axis paths in the old coned graph. Embedded geodesics map to uniform quasigeodesics with no repeated old apices (actual apex map injective). Recall the proper angle property at an old apex: for two incident ordinary vertices joined avoiding that apex in bounded length, the displacement belongs to a finite list inside the corresponding peripheral. Project a new cyclic coset coarsely to the quasiconvex embedded orbit of B in this old graph. If two projections are far apart in the old metric, a B -comparison quasigeodesic between them, except for bounded ends, stays boundedly near that axis (coarse projection and thin quadrilaterals). Frame vertices b sufficiently inside this portion are in fact at finite-list ordinary displacement from axis frame points a . Indeed nearby apex turns on the comparison path have bounded angle: bypass via two short bridges to the axis a sufficiently large fixed distance before and after, avoiding the apex by quasigeodesicity, and the bounded axis subpath. The short bridge at b itself then has bounded angles at each jump, using one farther such bridge for a bypass and replacing occurrences of that apex along the comparison path by the bounded-angle detour. Take bridges geodesic so do not revisit the apex. Proper angles give b^ -1 a from a fixed finite list. Pairs with a given displacement form one orbit under the coset stabilizer in B . Normalizing one pair shows there are finitely many coset types with this large projection, and all such middle projections are at bounded relative distance of one another after coning the stabilizer if nontrivial (otherwise already bounded; finitely many pair orbits in the normalized types). End errors are bounded too by the old relative embedding. Conversely every infinite cyclic intersection gives unbounded old projection. This gives exactly the finite induced new list (malnormal loxodromic cyclic subgroups also in B , by the old embedding), and the nearest projection now bounds each additional shortcut in the enlarged metric of B . Hence the assertions on relative hyperbolicity and embedding. Apply with B=Y_f . All these constructions use only finite orbit lists. "Full" relative maps of peripheral groups are not required.