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arxiv: 2412.03067 · v3 · pith:DS7UAWTOnew · submitted 2024-12-04 · 🧮 math.PR · math.GR· math.GT

Geodesic Trees and Exceptional Directions in FPP on Hyperbolic Groups

Riddhipratim Basu , Mahan Mj This is my paper

Pith reviewed 2026-05-23 08:00 UTC · model grok-4.3

classification 🧮 math.PR math.GRmath.GT
keywords first passage percolationhyperbolic groupsexceptional directionsgeodesic treesHausdorff dimensionPatterson-Sullivan measurecoalescenceboundary of hyperbolic group
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The pith

In first passage percolation on Gromov-hyperbolic groups, the exceptional directions where geodesics fail to coalesce or be unique almost surely form a set of strictly smaller Hausdorff dimension than the boundary.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies infinite geodesics in first passage percolation on Gromov-hyperbolic groups. Earlier work showed that for any fixed boundary point, geodesics from every vertex are almost surely unique and coalesce into a tree. Here the focus shifts to the random set of directions where uniqueness or coalescence fails. Under mild conditions on the passage-time distribution, this exceptional set almost surely has Hausdorff dimension strictly less than that of the boundary and hence carries Patterson-Sullivan measure zero. The authors also prove that, for groups that are not virtually free, such directions exist, are dense in the boundary, and become uncountable when the boundary has topological dimension greater than one.

Core claim

Under mild conditions on the passage-time distribution, the set of exceptional directions almost surely has strictly smaller Hausdorff dimension than the boundary of G and therefore has Patterson-Sullivan measure zero. For groups that are not virtually free, exceptional directions exist almost surely and are dense in the boundary; when the topological dimension of the boundary is n greater than 1, there exist directions admitting at least n+1 disjoint geodesics, and when the dimension exceeds one there are uncountably many exceptional directions. These conclusions rest on the structure of two random geodesic trees: the tree of all geodesics emanating from a fixed base point and the tree of a

What carries the argument

The two random geodesic trees (one rooted at a fixed vertex, one consisting of all semi-infinite geodesics in a fixed direction) whose coalescence failures define the exceptional set on the boundary.

If this is right

  • Exceptional directions carry Patterson-Sullivan measure zero.
  • For non-virtually-free groups, exceptional directions exist almost surely and are dense in the boundary.
  • When the topological dimension of the boundary exceeds one, there are uncountably many exceptional directions almost surely.
  • When the topological dimension equals n, there exist directions admitting at least n+1 disjoint geodesics almost surely.
  • An upper bound holds on the maximum number of disjoint geodesics sharing the same direction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same dimension-drop argument may apply to other coalescence questions on hyperbolic spaces once the relevant tree structure is established.
  • In groups with boundary dimension one, the exceptional set could still be dense yet countable, consistent with the stated results.
  • The upper bound on the number of disjoint geodesics supplies a uniform control that could be used to compare different passage-time distributions.

Load-bearing premise

Deep, unstated structural facts about Gromov-hyperbolic groups and their boundaries suffice to control the geometry of the random geodesic trees and the measures on the boundary.

What would settle it

A concrete passage-time distribution on a specific hyperbolic group for which the exceptional set has Hausdorff dimension equal to that of the boundary.

Figures

Figures reproduced from arXiv: 2412.03067 by Mahan Mj, Riddhipratim Basu.

Figure 1
Figure 1. Figure 1: Proof of Lemma 2.3: if the event AR described in the proof of the lemma is to hold, the FPP length of the red path restricted between u1 and v1 must be smaller than the blue path [u1, u2]∪[u2, v2]∪[v2, v1]. Since the word length of the red path is much larger than that of the blue path, this is sufficiently unlikely so that we can take a union bound over all possible locations of u1, v1 and w. in [u, v]ω ∩… view at source ↗
Figure 2
Figure 2. Figure 2: Proof of (1): We consider the hyperplanes Hi per￾pendicular to [1, v] with separation D between consecutive hyper￾planes. If there exist points v1, v2 within distance εn of v such that [1, v1]ω and [1, v2]ω are edge disjoint then there must be disjoint geodesics from Hi to Hi+1 for all i ≤ ε ′n. Since the probability of this event is bounded away from 1, and the events are independent for all even i, we ge… view at source ↗
read the original abstract

We continue the study of the geometry of infinite geodesics in first passage percolation (FPP) on Gromov-hyperbolic groups G, initiated by Benjamini-Tessera and developed further by the authors. It was shown earlier by the authors that, given any fixed direction $\xi\in \partial G$, and under mild conditions on the passage time distribution, there exists almost surely a unique semi-infinite FPP geodesic from each $v\in G$ to $\xi$. Also, these geodesics coalesce to form a tree. Our main topic of study is the set of (random) exceptional directions for which uniqueness or coalescence fails. We study these directions in the context of two random geodesics trees: one formed by the union of all geodesics starting at a given base point, and the other formed by the union of all semi-infinite geodesics in a given direction $\xi\in \partial G$. We show that, under mild conditions, the set of exceptional directions almost surely has a strictly smaller Hausdorff dimension than the boundary, and hence has measure zero with respect to the Patterson-Sullivan measure. We also establish an upper bound on the maximum number of disjoint geodesics in the same direction. For groups that are not virtually free, we show that almost surely exceptional directions exist and are dense in $\partial G$. When the topological dimension of $\partial G$ is greater than one, we establish the existence of uncountably many exceptional directions. When the topological dimension of $\partial G$ is $n$, we prove the existence of directions $\xi$ with at least $(n+1)$ disjoint geodesics. Our results hinge on deep facts about hyperbolic groups. En route, we also establish facts about the structure of random bigeodesics that substantially strengthen prior results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper continues the study of infinite geodesics in first-passage percolation on Gromov-hyperbolic groups. Building on prior results that unique semi-infinite geodesics exist to any fixed boundary point ξ and coalesce into a tree, it analyzes the random set of exceptional directions where uniqueness or coalescence fails. Using two random geodesic trees (one rooted at a basepoint and one consisting of all semi-infinite geodesics in a fixed direction), the authors claim that under mild conditions on the passage-time distribution this exceptional set has strictly smaller Hausdorff dimension than the boundary almost surely (hence Patterson-Sullivan measure zero), obtain an upper bound on the maximum number of disjoint geodesics sharing a direction, prove existence and density of exceptional directions when G is not virtually free, and establish multiplicity results when the topological dimension of ∂G exceeds 1. All results are stated to hinge on deep facts about hyperbolic groups and their boundaries; additional facts about random bigeodesics are obtained en route.

Significance. If the dimension-drop and multiplicity claims hold, the work supplies quantitative control on the 'bad' set of directions in FPP on hyperbolic groups, extending the authors' earlier uniqueness and coalescence theorems. The explicit dependence on the topological dimension of the boundary and the measure-zero conclusion with respect to the Patterson-Sullivan measure are potentially useful for further geometric-probabilistic analysis. The strengthening of bigeodesic results is a secondary contribution.

major comments (2)
  1. [Abstract] Abstract: the central claim that the exceptional set has strictly smaller Hausdorff dimension than ∂G (and is therefore Patterson-Sullivan null) is asserted to follow from 'deep facts about hyperbolic groups,' yet no specific theorems (e.g., on the geodesic flow, conformal measures, or boundary structure) are identified, nor is the reduction from the random trees to those facts supplied. This step is load-bearing for the dimension inequality and cannot be verified from the given statement.
  2. [Abstract] Abstract: the new dimension and multiplicity statements are described as building on the authors' prior papers, but the abstract gives no indication that the strict Hausdorff-dimension drop reduces to quantities already controlled in those works; without an explicit reduction the derivation risks circularity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and for identifying points where the abstract could more clearly articulate the logical dependencies. We address each major comment below. Revisions will be made to the abstract to improve transparency without changing the paper's core arguments or results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the exceptional set has strictly smaller Hausdorff dimension than ∂G (and is therefore Patterson-Sullivan null) is asserted to follow from 'deep facts about hyperbolic groups,' yet no specific theorems (e.g., on the geodesic flow, conformal measures, or boundary structure) are identified, nor is the reduction from the random trees to those facts supplied. This step is load-bearing for the dimension inequality and cannot be verified from the given statement.

    Authors: We agree that the abstract is too terse on this point. The dimension inequality is obtained in the body by combining the almost-sure tree structure and coalescence properties of the two random geodesic trees (from our prior uniqueness/coalescence theorems) with deterministic facts on hyperbolic boundaries: specifically, the Ahlfors regularity of the Patterson-Sullivan measure and the fact that certain subsets defined by multiple geodesics have strictly lower Hausdorff dimension. The reduction is therefore non-circular and rests on standard references (e.g., results on conformal measures and the geodesic flow). We will revise the abstract to name these facts explicitly and to indicate where the reduction occurs. revision: yes

  2. Referee: [Abstract] Abstract: the new dimension and multiplicity statements are described as building on the authors' prior papers, but the abstract gives no indication that the strict Hausdorff-dimension drop reduces to quantities already controlled in those works; without an explicit reduction the derivation risks circularity.

    Authors: The argument is not circular. The probabilistic control (uniqueness to each ξ, coalescence into trees, and the resulting exceptional set) is taken from our earlier papers. The strict Hausdorff-dimension drop is then a deterministic consequence of applying those controls to the boundary geometry of G (using that the boundary is Ahlfors regular and that the exceptional set is a countable union of lower-dimensional subsets). We will revise the abstract to state this separation of probabilistic and deterministic ingredients explicitly, thereby removing any appearance of circularity. revision: yes

Circularity Check

1 steps flagged

Minor self-citations to authors' prior results on geodesic uniqueness, but new Hausdorff dimension claims on exceptional directions do not reduce to those inputs.

specific steps
  1. self citation load bearing [Abstract]
    "It was shown earlier by the authors that, given any fixed direction ξ∈∂G, and under mild conditions on the passage time distribution, there exists almost surely a unique semi-infinite FPP geodesic from each v∈G to ξ. Also, these geodesics coalesce to form a tree."

    The uniqueness and tree formation for fixed directions are taken from the authors' own prior work; the new analysis of the random set of exceptional directions (where these fail) builds directly on this without an independent derivation of the dimension drop from first principles or external theorems alone.

full rationale

The paper explicitly continues prior work by the same authors on uniqueness and coalescence for fixed directions, citing those results as background. However, the central claims (strictly smaller Hausdorff dimension of the exceptional set, multiplicity bounds, density) are new and do not reduce by any stated equation or definition to quantities fitted or defined in the cited prior papers. The mention of 'deep facts about hyperbolic groups' is acknowledged as external input rather than a self-referential loop. This matches the pattern of normal continuation with one minor self-citation that is not load-bearing for the novel results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard facts from the theory of Gromov-hyperbolic groups (boundaries, Patterson-Sullivan measures, and geodesic properties) together with mild assumptions on passage-time distributions; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption Gromov-hyperbolic groups possess boundaries equipped with Patterson-Sullivan measures that control dimension and measure-zero sets
    Invoked to conclude that smaller Hausdorff dimension implies Patterson-Sullivan measure zero.
  • domain assumption Mild conditions on the passage-time distribution guarantee existence and coalescence of semi-infinite geodesics to a fixed boundary point
    Stated as the setting under which the exceptional-set results hold.

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