Quantitative hyperbolic geometry toolkit

The following toolkit holds, with each clause under its own stated hypotheses. Write $M(\lambda ,\epsilon ,\delta )=92{\lambda }^2(\epsilon +3\delta )$. Assume AC for the Morse projection families, selection of a coarse inverse, linear-filling converse, general boundary extension and proper boundary compactness. The elementary metric clauses, finite-word and group-orbit clauses, sequence-product topology and loxodromic dynamics below are choice-free; inverse estimates for an already supplied selector are also choice-free.

1. A connected cycle-free unit-edge graph has unique geodesics and tripod triangles, hence is $0$-slim. In any geodesic space, $\delta$-slimness implies product constant $3\delta$. In any metric space, product constant $\kappa$ at every basepoint is equivalent to the four-point condition with largest two opposite-pair sums differing by at most $2\kappa$. In a geodesic space that condition implies $4\kappa$-slimness.

2. In a geodesic $\delta$-slim space, for $\delta >0$, a $k$-local arc-length geodesic with $k\ge 6\delta$ is a $(3,4\delta )$-quasi-geodesic. When $\delta =0$, every positive-locality geodesic is globally geodesic. Under AC, every possibly discontinuous $(\lambda ,\epsilon )$-quasi-geodesic on a nonempty compact real interval in such a space has Hausdorff distance at most $M(\lambda ,\epsilon ,\delta )$ from every specified endpoint geodesic. Properness is unnecessary.

3. For a $(\lambda ,\epsilon )$ embedding $f:X\to Y$ with attained coarse-density radius $R$, AC supplies an inverse selector $g$ with $d_Y(fg(y),y)\le R$, $d_X(gf(x),x)\le \lambda (R+\epsilon )$ and embedding constants $(\lambda ,\lambda (2R+\epsilon ))$; these estimates are choice-free if the selector is supplied, and do not require geodesicity. Under AC, if $X,Y$ are geodesic and $Y$ is $\delta$-slim, the embedding $f$ alone implies that $X$ is $\lambda (2M(\lambda ,\epsilon ,\delta )+\delta +\epsilon )$-slim. Consequently quasi-isometric geodesic spaces share hyperbolicity.

4. In the standing finitely generated hyperbolic-group convention, for any integer $\mathrm{\Delta }\ge \max \{3,2\delta \}$ all formal null words of length at most $10\mathrm{\Delta }$ form a finite presentation. Every nonempty freely reduced null word has a based contiguous subword $u$ with a strictly shorter replacement $v$ and relator spelling $u{v}^{-1}$ in this set, so $u$ is more than half the spelling. The cyclic version, allowing basepoint crossing and free cyclic reduction, holds as well. Relator spellings need not be freely reduced.

5. Conversely, under AC, a finite presentation with relator lengths at most $L\ge 0$ and algebraic area $\mathrm{Area}(w)\le K\mathrm{\mid }w\mathrm{\mid }$ for every null word has uniformly slim triangles and thin bigons in the labelled unit-edge Cayley realization, with a constant depending only on $K,L$. If ${\delta }_0(K,L)$ denotes the common simple-realization bound from the filling supplier, $12{\delta }_0(K,L)+8$ is a labelled-realization bound. No numerical formula for ${\delta }_0$ is claimed.

6. Every infinite-order element $g$ of a finitely generated hyperbolic group has a quasi-isometrically embedded power orbit and ${\tau }_S(g)>0$; in fact $\mathrm{\mid }{g}^n{\mathrm{\mid }}_S\ge \mathrm{\mid }n\mathrm{\mid }\mathrm{/}C$ and ${\tau }_S(g)\ge 1\mathrm{/}C$ for one positive integer $C$ depending only on the specified Cayley graph and slimness. Its centralizer has $⟨g⟩$ of finite index, as does the subgroup preserving its unordered pole pair. The full orbit-chord and coset bounds are retained: with orbit constants $\lambda ,c$, $\kappa =3\delta$, and an integer $J\ge 4$ satisfying $2^J\mathrm{/}\lambda -c>\lambda +c+4\kappa J$, there is a power of two $N\le 2^{J-1}$ for which $x_i=g^{iN}$, $L=d(e,x_1)$, $K=L-d(e,x_2)\mathrm{/}2\ge 0$, $L>2K+4\kappa$ and $L\le \lambda 2^J+c$. Put $E=K+2\kappa +3\delta +1$. Every finite chain $x_a,\dots ,x_b$ and every specified endpoint chord have the two distance bounds $<E$ from chain to chord and $<L+3E$ from chord to chain. Every right or left coset of $⟨g⟩$ in the pole-pair stabilizer, and hence in the centralizer, has a representative of length less than $L+4K+10\kappa +12\delta +5\le 5(\lambda 2^J+c)+10\kappa +12\delta +5$.

7. In a product-$\kappa$ metric space, joint mixed-product divergence is an equivalence relation on Gromov sequences, independent of basepoint. For representative product $P$ and supremal boundary product $B$, $B-2\kappa \le P\le B$ when finite, and infinite value is exactly equality of classes. A basepoint change by distance $D$ changes products by at most $D$; boundary products obey the minimum inequality with loss $3\kappa$. The criterion that every point of an open set contains some threshold neighbourhood $\{\eta :B(\xi ,\eta )>R\}$ defines a Hausdorff topology, independent of basepoint and supplied representatives; threshold neighbourhoods need not themselves be open. Under AC, a quasi-isometry of geodesic hyperbolic spaces induces a homeomorphism of these sequence boundaries, bounded-distance maps induce the same map, and extensions respect composition. Properness is not imposed here.

8. Under AC, for a nonempty proper geodesic product-$\kappa$ space, every sequence-boundary class is represented by a geodesic ray from any fixed basepoint. The quotient of those rays by finite Hausdorff distance, with the topology induced by uniform convergence on bounded parameter intervals, is homeomorphic to the sequence boundary. The boundary is compact, including when empty.

9. A finitely generated hyperbolic group that is neither finite nor virtually cyclic has independent infinite-order elements. More precisely, intersecting pole sets of two infinite-order elements are equal; after orienting their positive poles to agree, some positive powers are equal. Independent loxodromics have four pairwise disjoint open pole neighbourhoods. Every loxodromic isometry of a product-hyperbolic metric space, and in particular every infinite-order element of the standing hyperbolic group, has uniform north–south dynamics: for neighbourhoods $U_{+},U_{-}$ of its poles, some integer $n_0$ satisfies $g^n(\mathrm{\partial }X\setminus U_{-})\subseteq U_{+}$ and $g^{-n}(\mathrm{\partial }X\setminus U_{+})\subseteq U_{-}$ for all $n\ge n_0$.