Strongly Bolic Metric

• Strongly bolic metrics are refined metric structures that enhance Gromov hyperbolicity through stringent smoothness and convexity conditions.
• They provide precise control over four-point configurations and midpoint stability, enabling detailed analyses of Ptolemy spaces and relatively hyperbolic groups.
• Their framework supports applications in rigidity theory and the Baum–Connes conjecture via explicit analytic estimates and group-invariant metric constructions.

A strongly bolic metric is a metric structure introduced to provide precise analytic control over the geometry of spaces and groups, strengthening the classical notion of Gromov hyperbolicity and Lafforgue’s bolicity property. These metrics satisfy robust smoothness and convexity conditions, which ensure the “stability” of midpoints and four-point configurations, allowing for applications in rigidity, higher index theory, and the Baum–Connes conjecture. Strong bolicity plays a central role in the analysis of relatively hyperbolic groups, Ptolemy spaces, and spaces with controlled geometric and analytic properties (Lajoinie-Dodel, 24 Dec 2025, Xiao et al., 2018).

1. Foundational Definitions

The concept of strong bolicity builds on refinement of geodesicity and local finiteness. A metric space $(X,d)$ is called $\eta$-weakly geodesic if for any $x, y \in X$ and any $t \in [0, d(x, y)]$, there is $z \in X$ such that $d(x, z) \leq t + \eta$ and $d(z, y) \leq d(x, y) - t + \eta$. Uniform local finiteness requires boundedness on the size of metric balls.

The axiomatic definition of a strongly bolic metric, as formalized in the context of Kasparov–Skandalis and Lafforgue's work and in (Lajoinie-Dodel, 24 Dec 2025), requires:

• The existence of constants $\eta, r > 0$ and a midpoint map $m: X \times X \to X$ such that:
  • (strong-$B_1$) Smoothness: For any $\rho > 0$ and $\eta' > 0$, there exists $R$ so that for $d(x_1, x_2) \leq \rho$, $d(y_1, y_2) \leq \rho$, and $d(x_i, y_j) \geq R$, the four-point estimate

  $$| d(x_1, y_1) + d(x_2, y_2) - d(x_1, y_2) - d(x_2, y_1) | \leq \eta'$$

  holds. - (weak-$B_2'$) Convexity: There is $\eta_1 > 0$ such that for all $x, y, z \in X$,

  $$|2\,d(x, m(x, y)) - d(x, y)| \leq 2\eta_1,\quad |2\,d(y, m(x, y)) - d(x, y)| \leq 2\eta_1,$$

  $$d(m(x, y), z) \leq \max\{d(x, z), d(y, z)\} + 2\eta_1.$$

  Furthermore, $m$ “pushes points” away from far-off third points: for every $p > 0$ there is $N(p)$ such that if $d(x, z), d(y, z) \leq N$ and $d(x, y) > N$, then $d(m(x, y), z) \leq N - p$.

Strong bolicity is designed to be a uniform, quantitative strengthening of Gromov hyperbolicity, adapted to applications where analytic and geometric stability are required.

2. Strong Bolicity and Relatively Hyperbolic Groups

Strongly bolic metrics play a decisive role in understanding the geometry of relatively hyperbolic groups. For a finitely generated group $G$ hyperbolic relative to subgroups $P_1,\ldots,P_k$, a metric $\hat{d}$ is constructed that is $G$-invariant, quasi-isometric to the word metric $d_G$, uniformly locally finite, $\eta$-weakly geodesic, and strongly bolic (Lajoinie-Dodel, 24 Dec 2025).

The construction uses the coned-off Cayley graph $$X_c = \widehat{\Gamma} = \widehat{\mathrm{Cay}}(G, S)$$, whose vertex set splits as $G \cup X_c^\infty$, where $X_c^\infty$ corresponds to cone-points for cosets $gP_i$. To build $\hat{d}$:

• For $a \in G$, the per-vertex pseudo-metric $d_a(x, y)$ is an indicator of geodesic containment.
• For $a \in X_c^\infty$, $d_a(x, y)$ is defined via the $L^1$-distance or a strongly bolic metric $d_b$ on probability measures supported in $gP_i$, using structures called masks (see below).
• The metric is then assembled as $D(x, y) = \sum_{a \in X_c} \theta(d_a(x, y))$ for an appropriately chosen bump function $\theta$, yielding $\hat{d}(x, y) = D(x, y) + C_\triangle$ for $x \neq y$.

This construction equips $G$ with a metric environment suitable for the analysis of rigidity and higher-index phenomena.

3. Masks, Pseudometrics, and Analytic Estimates

Masks, introduced by Chatterji and Dahmani, are probability measures $\mu_x(a)$ associated to each vertex $a \in X_c$ and group element $x \in G$, obtained by a canonical “flow” along cone geodesics. These masks stabilize after finitely many steps and localize the support near $a$.

Key analytic estimates for masks include:

• For adjacent $x, y$, $$\sum_{a \in X_c} \|\mu_x(a) - \mu_y(a)\|_1^p < \infty$$ for suitable $p$.
• There is exponential decay: $$\|\mu_x(a) - \mu_y(a)\|_1 \leq \kappa^{d_c(a,x) + \Theta(a,x)}$$ with $\kappa \in (0,1)$ and $\Theta(a,x)$ a sum of angle terms.

These estimates support the control of the assembled metric $\hat{d}$ and the verification of the strong-$B_1$ four-point estimate and weak-$B_2'$ convexity conditions (Lajoinie-Dodel, 24 Dec 2025).

4. Strongly Bolic Metrics in Ptolemy and Metric Spaces

Strongly bolic metrics are closely related to strong hyperbolicity in the sense of Nica–Špakula and Lafforgue. In Ptolemy spaces (metric spaces satisfying the Ptolemy inequality), several explicit constructions yield strongly hyperbolic (and hence strongly bolic) metrics (Xiao et al., 2018):

• Log-metric: $\rho(x, y) = \log(1 + d(x, y))$, which satisfies a four-point exponential inequality and is strongly hyperbolic with parameter $\varepsilon = 2$.
• Scale-invariant and inversion-type metrics: These are modifications involving normalization by basepoints or powers of distances, each verified to be strongly hyperbolic and Gromov hyperbolic.

A space that is strongly hyperbolic with parameter $\varepsilon$ is Gromov-hyperbolic with $\delta = (\ln 2)/\varepsilon$, and if it is roughly geodesic, it is strongly bolic (Xiao et al., 2018).

5. Applications: Baum–Connes Conjecture and Admissible Parabolics

For groups $G$ hyperbolic relative to admissible parabolic subgroups such as proper cocompact $CAT(0)$-groups or virtually abelian groups, the constructed metric $\hat{d}$ allows the application of Lafforgue’s results on the Baum–Connes conjecture. If each parabolic has the rapid decay property (RD), $G$ satisfies the Baum–Connes conjecture (Lajoinie-Dodel, 24 Dec 2025).

Admissibility entails that $\Prob_C(P)$, the space of probability measures of bounded support in a parabolic subgroup $P$, carries a canonical strongly bolic metric coming from barycenters in $CAT(0)$ spaces and the Wasserstein distance.

6. Examples, Characterizations, and Further Properties

The framework unifies the analysis of many classical spaces:

• Euclidean spaces and Ptolemy subspaces can be endowed with strongly hyperbolic metrics; in particular, their log-metrics turn them into strongly Gromov-hyperbolic spaces, even when their original metrics are not.
• Möbius maps between punctured domains act as quasi-isometries for strong bolic metrics such as inversion-metrics (Xiao et al., 2018).

Key properties of strongly bolic metrics in these constructions include invariance under quasi-isometries (in group actions), uniform local finiteness, and the existence of explicit midpoint maps and analytic bounds on configuration stability.

7. Theoretical Implications and Connections

Strong bolicity provides an analytic framework extending traditional Gromov hyperbolicity, supporting robust four-point control and midpoint stability for large-scale geometry and harmonic analysis. Its development is central to rigidity conjectures, higher index theory, and the interplay between group-theoretic, metric, and operator-algebraic invariants. The notion also underpins effective transfer of sofic rigidity and works as a criterion for group actions to admit transfer via rigid-analytic techniques, notably in the context of the Baum–Connes conjecture for relatively hyperbolic groups with admissible parabolics (Lajoinie-Dodel, 24 Dec 2025, Xiao et al., 2018).