Random quotients preserve acylindrical and hierarchical hyperbolicity

TL;DR

The paper develops a probabilistic framework for random quotients of acylindrically and hierarchically hyperbolic groups by taking kernels of independent random walks. It proves that quotients by these random subgroups are, with probability tending to 1, still acylindrically hyperbolic, and preserves hierarchical hyperbolicity in the HHG setting (including relative HHGs). The key techniques introduce spinning families and projection complexes, along with a modified cone-off and minimal-representative machinery to lift quotient structures back to the parent group. The results extend to relative and common-quotient contexts, and yield exotic HHG quotients with fixed-point properties like Kazhdan’s property (T). These methods provide a robust toolkit for producing negatively curved-like quotients and show HHGs and AHGs are abundant under random quotients. Overall, the work broadens the landscape of quotients that retain strong hyperbolic and hierarchical structures, with implications for rigidity and fixed-point phenomena in geometric group theory.

Abstract

We propose a new model for random quotients of groups using independent random walks. In this model, we show that random quotients of acylindrical hyperbolic groups asymptotically almost surely remain acylindrically hyperbolic. Our main tools relate the theories of spinning families and projection complexes to random walks. In the presence of a hierarchical hyperbolic structure on the group, we leverage the fine control of projections to show that this structure is preserved in the quotient asymptotically almost surely. The same techniques yield that random quotients of a non-elementary hyperbolic group (relative to any finite collection of finitely generated peripheral subgroups) are asymptotically almost surely hyperbolic (relative to commensurable peripheral subgroups). Finally, we also prove that any two groups that are both acylindrically and hierarchically hyperbolic have a common quotients which is itself acylindrically and hierarchically hyperbolic. This produces "exotic" hierarchically hyperbolic groups with strong fixed point properties, such as Kazhdan's property (T).

Figures (17)

• Figure 1: The domains and geodesic involved in the proof of \ref{['projaxiom:finitelymanybig']} in \ref{['P:hyp-projcplx']}. The geodesic $[u,v]$ has a long subsegment in a uniform neighborhood of $Y$, and the endpoints $a$ and $b$ of this subsegment project to $Y$ close to the endpoints of the whole geodesic.
• Figure 2: The open lift from the proof of \ref{['prop:liftingQuadrilaterals']}, and its bending at the shortening vertex (here, the dashed line).
• Figure 3: Proof of \ref{['lem:bgi_with_U']}, in the case $z'$ does not lie on $\gamma$.
• Figure 4: The geodesics from the proof of \ref{['prop:LiftsProjCloseInU']}. After bending $\gamma$ and $\gamma'$ a finite number of times, their endpoints coincide. Notice that $\gamma$ and $\gamma_k$ must overlap on the segment of $\gamma$ connecting $v_U$ to the first vertex of the form $v_Y$ for some $Y\in \mathcal{Y}$, as bending can occur only at cone points.
• Figure 5: The various lifts from the proof of \ref{['lem:lifting_trans_and_points']} (for $k=3$).

Theorems & Definitions (165)

• Theorem A
• Theorem B
• Corollary C
• Corollary D
• Theorem E
• Corollary F
• Corollary G
• Theorem H
• Theorem I
• Lemma 2.1