On the connectedness of the boundary of hierarchically hyperbolic spaces

Ravi Tomar

REVIEW 4 major objections 5 minor 24 references

On the connectedness of the boundary of hierarchically hyperbolic spaces

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves that a weakly visible boundary of a one-ended proper geodesic metric space is connected, and that a hierarchically hyperbolic group has connected hierarchical boundary exactly when it is one-ended.

desk verdict The DHS conjecture resolution looks essentially right, but the free-product application rests on a genuinely unproven lemma. read the letter →

arxiv 2509.00321 v1 pith:3CZ7WLSP submitted 2025-08-30 math.GR

Ravi Tomar This is my paper · ORCID

classification math.GR MSC 20F6520F67

keywords hierarchicallyhyperbolicgroupshierarchicalboundaryone-endedweaklyvisibleconnectedfreeproductsBass–Serretreemappingclassgroup

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that the hierarchical boundary of a one-ended hierarchically hyperbolic group is connected, and that, conversely, a hierarchically hyperbolic group with connected hierarchical boundary must be one-ended. The forward direction follows from a general topological theorem: every weakly visible boundary of a one-ended proper geodesic metric space is connected. The equivalence settles an open conjecture stated in [10]. As an application, the paper shows that for free products of one-ended hierarchically hyperbolic groups, the homeomorphism type of the boundary of the product is determined by, and determines, the homeomorphism types of the factor boundaries. This matters because the hierarchical boundary is the natural compact boundary for a broad class of groups including mapping class groups, and connectedness of such boundaries is a known way to detect one-endedness.

What carries the argument

The load-bearing device is the notion of a weakly visible boundary: a boundary of a proper geodesic metric space in which two uniformly bounded sequences that converge in the compactification converge to the same boundary point. The proof of Theorem 3.1 uses one-endedness to find geodesic segments joining points near two hypothetical boundary components while staying outside arbitrarily large balls; weak visibility then forces the two components to share a boundary point, a contradiction. For the group-theoretic converse, the machinery is the end classification of finitely generated groups, the splitting of infinite-ended groups over finite subgroups, and the quotient map from the hierarchic

What would settle it

The most direct test is computational: for G = A * B with A and B one-ended hierarchically hyperbolic groups, construct the boundary using the Section 4 structure and check whether it has more than one connected component—specifically, the factor boundaries attached to the Bass–Serre tree's vertices should be isolated components. If this boundary is connected, Proposition 6.1 and Theorem 6.3 fail. A second test targets Theorem 3.1: any one-ended proper geodesic metric space with a weakly visible boundary should have connected boundary; exhibiting such a space whose boundary admits a separation

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.6: a hierarchically hyperbolic group has a connected hierarchical boundary exactly when the group is one-ended. The hierarchical boundary, introduced in [10], is a compact metrizable boundary built from weighted sums of points in the Gromov boundaries of the hyperbolic spaces appearing in the HHS structure. The forward direction follows from Theorem 3.1, which says any weakly visible boundary of a one-ended proper geodesic metric space is connected; hierarchical boundaries are weakly visible. The converse uses the classification of ends: two-ended groups are virtually cyclic with two-point boundary, while infinite-ended groups split over finite subgroups and ha

Load-bearing premise

The converse of Theorem 3.6 assumes that an infinite-ended hierarchically hyperbolic group splitting over finite subgroups is hyperbolic relative to its infinite vertex groups in a way compatible with its hierarchical structure, so that the known quotient map onto the Bowditch boundary applies.

Editorial extensions

If this is right

The hierarchical boundary of the mapping class group of a connected orientable surface of finite type with 3g+n-3 at least 2 is connected.
Any hierarchically hyperbolic group with a connected hierarchical boundary is one-ended; in particular, it cannot be two-ended and cannot be an infinite-ended group admitting the relative-hyperbolic quotient used in the proof.
For free products of one-ended hierarchically hyperbolic groups with the Section 4 hierarchical structure, the homeomorphism type of the boundary of the product is completely determined by the homeomorphism types of the factor boundaries, and conversely.
The free-product construction gives a way to compare two hierarchical structures on the same one-ended group: form the free product of the group with itself and compare the boundaries of the two products.
Free products of locally hierarchically quasiconvex hierarchically hyperbolic groups are again locally hierarchically quasiconvex with respect to the constructed structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The weakly visible boundary theorem is stated in maximal generality, so it should apply to other coarse compactifications that satisfy the same uniform-convergence condition; one concrete candidate is a class of Floyd boundaries with a suitable visibility axiom.
The component structure of the free-product boundary suggests a wider rigidity pattern: for any graph of groups with one-ended vertex stabilizers, the connected components of the hierarchical boundary may encode exactly the vertex stabilizers of the Bass–Serre tree, making the splitting readable from the boundary.
The connectedness results for the hierarchical boundary and for the Z-boundary of the mapping class group are consistent with the possibility that the two compactifications are homeomorphic, though the paper gives no direct evidence for that stronger statement.
The paper's closing suspicion that locally hierarchically quasiconvex HHGs are hyperbolic and locally quasiconvex, if pursued, would imply that the new combination theorem for locally HQC groups is only nontrivial in the non-hyperbolic regime; this could be tested by classifying locally HQC HHGs that are not hyperbolic.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my read. The core result, Theorem 3.6, is the right kind of thing: it gives the clean iff characterization of connectedness of the hierarchical boundary in terms of one-endedness for HHGs, and it resolves the Durham–Hagen–Sisto conjecture. Theorem 3.1, the general weak-visibility statement for compactifications of one-ended proper geodesic spaces, is a nice unifying observation, and it feeds the mapping class group boundary and the Hamenstädt Z-boundary. The paper is honestly written and the central argument is plausible.

The soft spots are real but mostly repairable. In Theorem 3.1, the proof claims one-endedness gives geodesics outside a compact set; one-endedness actually gives paths outside the compact set, not necessarily geodesics. The argument goes through with paths, so this is a minor fix. In the converse of Theorem 3.6, the author invokes a splitting over finite edge groups and then relative hyperbolicity, and applies [3, Thm 1.3] to get a quotient map from the hierarchical boundary to the Bowditch boundary. The compatibility of the HHG structure with the relative hyperbolicity is not checked. That might be fine if [3] applies in full generality, but it needs to be stated and justified.

The genuinely load-bearing gap is in Lemma 5.1, the free-product compactification lemma. The back-and-forth construction of f: G1 -> G2 does not give the control needed for continuity at boundary points. Step 1 assigns f(g_k) with d(f(g_k), Bf(pi(g_k))) < 1/k, but Step 2, which exists to make f surjective, only ensures d(h_k, Bf(pi(g))) < d(h_k, BG2) + 1/k. Since d(h_k, BG2) is a fixed positive number for a given group element, this does not tend to zero, and the proof's assertion that d(Bf(pi(g_k)), f(g_k)) -> 0 is unjustified for elements assigned in Step 2. The continuity of f ∪ Bf is not established, and since Lemma 5.1 feeds the Bass–Serre tree isomorphism in Theorem 5.3 and the rigidity theorem 6.3, that whole application is currently unproven.

Bottom line: the main connectedness theorem is likely correct and worth a referee's time. The free product rigidity theorem needs a repaired Lemma 5.1 or a different argument. I'd send it to peer review and ask the author to fix the lemma and to spell out the compatibility issue in Theorem 3.6.

Referee Report

4 major / 5 minor

Summary. The paper proves three main results: (1) Theorem 3.1: every weakly visible metrizable compactification of a one-ended proper geodesic metric space has connected boundary; (2) Theorem 3.6: for a hierarchically hyperbolic group, the hierarchical boundary is connected if and only if the group is one-ended, resolving a conjecture of Durham–Hagen–Sisto; (3) Theorem 6.3: for free products of one-ended hierarchically hyperbolic groups, the hierarchical boundary of the free product determines the hierarchical boundaries of the factors, in the sense of a homeomorphism of the whole boundary exactly when the factor boundaries are pairwise homeomorphic. The proof of Theorem 3.1 is a direct compactness/weak-visibility argument. The free-product part builds a model compactification from a Bass–Serre tree and uses a back-and-forth lemma to promote boundary homeomorphisms to bijections of the vertex groups.

Significance. If the central claims are correct, the paper resolves an open conjecture about the connectedness of hierarchical boundaries, provides a clean general compactification theorem, and extends free-product rigidity results from Gromov/Bowditch/Morse/Floyd boundaries to the hierarchical setting. The proof of Theorem 3.1 is elementary and potentially widely applicable; the use of already-established theorems (e.g., the DHS boundary construction, the ABR quotient theorem) makes the argument largely non-circular and checkable. However, as detailed below, one of the stated applications (Theorem 6.3) is false without a permutation of factors, and two proofs contain load-bearing gaps that need repair.

major comments (4)

[§3, proof of Theorem 3.1] The proof asserts that one-endedness yields a sequence of geodesics γ_m joining subsequences of x_n and y_n such that γ_m is contained in X\K_m. This is not justified and is false in general: in R^2 with K_m a large disk, two points on opposite sides are joined by a Euclidean geodesic that crosses the disk, even though they lie in the same unbounded component of the complement. However, one-endedness does give a continuous path in X\K_m between suitably chosen points, and the later transition-point argument works for any continuous path. The proof should replace 'geodesics' with 'paths' and justify existence of the path from the one-endedness assumption.
[§6, Theorems 6.2 and 6.3] The 'only if' direction is false as stated because no permutation of factors is allowed. For example, let A_1=Z^2, B_1=Z^3, A_2=Z^3, B_2=Z^2. Then G_1=A_1*B_1 and G_2=B_1*A_1 are the same free product up to swapping the two factors, so their hierarchical boundaries are homeomorphic via that swap, but the conclusion '∂A_i ≅ ∂B_i for all i' would require ∂Z^2 ≅ ∂Z^3. The proof of Theorem 6.2 only shows that a homeomorphism of δ(Γ_1) and δ(Γ_2) induces a bijection between the non-singleton connected components, which gives a matching of factor boundaries only up to a permutation. The theorems should be restated with 'up to a permutation of the indices', and the proof should explicitly say that the homeomorphism induces such a permutation.
[§5, Lemma 5.1] The proof of continuity at boundary points is incomplete. For elements f(g_k) assigned in Step 2, the construction only gives d(f(g_k), Bf(π(g_k))) < d(f(g_k), BG_2) + 1/k, where the index k is the position of f(g_k) in the enumeration of G_2. The sentence 'From the definition of f, it follows that ... =0' silently assumes d(f(g_k),BG_2)->0. This is true for an enumeration of a discrete group because each h_k eventually leaves every finite set and therefore d(h_k,BG_2)->0, but this argument is absent. Since Lemma 5.1 underpins Theorem 5.3 and hence Theorem 6.3, the proof needs to spell this out.
[§3, proof of Theorem 3.6] The converse direction applies [7] to conclude that G is hyperbolic relative to the infinite vertex groups of a splitting over finite edge groups, and then applies Theorem 2.10 ([3, Theorem 1.3]) to pass to a quotient of the hierarchical boundary. The manuscript does not verify the compatibility hypotheses of [3, Theorem 1.3] (e.g., that the HHG structure is compatible with the relative hyperbolicity or with the peripheral subgroups). If that theorem requires such compatibility, the argument is incomplete. Also, the proof's trichotomy 'one-ended, two-ended, or infinite-ended' omits finite groups; a finite HHG has empty boundary, which is not connected, so this case should be mentioned explicitly.

minor comments (5)

[§3, Theorem 3.6 statement] The statement says 'Let (G,S) be an HHS' but the result is about hierarchically hyperbolic groups; it should say 'HHG'.
[§5, Theorem 5.3 notation] In the paragraph before Theorem 5.3, the notation line says 'Let Bf_1 : BA_1 → BA_1' but it should be 'Bf_1 : BA_1 → BA_2'.
[§4, Proposition 4.3] The proof of ψ being a homeomorphism is quite compressed. Several assertions such as 'from the definition of neighborhoods and the hierarchical structure of Γ, it follows that ...' are not demonstrated; since this proposition identifies the two compactifications, more detail would improve verifiability.
[§3, Lemma 3.2] The verification that Hamenstädt's Z-boundary is weakly visible is very brief and relies on [4, Lemma 3.19] without stating exactly which conditions (2) and (3) of [13, Definition 4.2] are being checked. A few more sentences would help.
[§4.2, relative projections] The definition of the relative projections ρ^V_U for U,V in different vertex spaces is terse. In particular, the notation ρ^U_hatΓ and the map sending a cone-point to ρ^{S_w}_U should be clarified, since these projections are used in verifying the HHS axioms.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation; self-citations are contextual only.

full rationale

The paper's central derivation chain is self-contained and does not reduce to its own inputs by construction. Theorem 3.1 proves that any weakly visible boundary of a one-ended proper geodesic metric space is connected directly from the definition of weak visibility and one-endedness; no fitted parameters or self-referential definitions are used. The connectedness of hierarchical boundaries (Corollary 3.4) is an immediate consequence of the known weak visibility of the HHS boundary [4, Lemma 3.20], an external cited result. Theorem 3.6's converse relies on standard theorems on ends and splittings ([8]), relative hyperbolicity ([7]), and the external quotient theorem [3, Theorem 1.3]; it does not invoke any result of the present author. The free-product application (Theorems 5.3, 6.3) is modeled on Martin–Świątkowski [15] and uses Lemma 5.1, whose proof is an explicit construction rather than a backward definition; even if that lemma contains a proof gap (as a skeptical reading might suggest), the gap is a matter of correctness, not circularity. The only self-citations are [22] and [9] by the author and coauthor; they appear solely in the introduction as context for analogous known results and are not used as evidence or premises in the proofs. The paper also skips routine verification of a neighborhood basis and refers to [14, Theorem 6.17], and omits the proof of Proposition 6.1 with a reference to [15]; these are omitted proofs, not circular steps. Overall, no claim is equivalent to its own assumption, no prediction is a renamed input, and no load-bearing argument reduces to a self-citation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces the definition of weakly visible boundary, but this is a property of existing boundaries, not a new entity. All substantive assumptions are standard results from the geometric group theory literature. The most fragile assumption is the compatibility of the HHG structure with the relative hyperbolicity used in Theorem 3.6.

assumptions (5)

domain assumption One-endedness: for every compact K in a proper geodesic metric space X, X\K has exactly one unbounded component.
Used in the proof of Theorem 3.1 to find paths between unbounded sequences avoiding compact balls.
standard math Stallings' splitting theorem: an infinitely-ended finitely generated group splits over finite subgroups.
Used in Case 2 of Theorem 3.6 to obtain a graph of groups decomposition.
domain assumption A graph of groups with finite edge groups and infinite vertex groups is hyperbolic relative to the vertex groups.
Used in Theorem 3.6 to invoke [3, Theorem 1.3] and the disconnection of the Bowditch boundary.
domain assumption The hierarchical boundary of a proper HHS is weakly visible ([4, Lemma 3.20]).
Needed to apply Theorem 3.1 to HHS boundaries (Corollary 3.4).
domain assumption For a hyperbolic group, the hierarchical boundary is homeomorphic to the Gromov boundary ([10, Theorem 4.3]).
Used in Theorem 3.6 Case 1 and the all-finite-vertex case to identify the boundary with a Cantor set.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the connectedness of the boundary of hierarchically hyperbolic spaces." pith.science (2026). https://pith.science/paper/3CZ7WLSP

@misc{pith2026250900321,
  author       = {Pith},
  title        = {Pith review of: On the connectedness of the boundary of hierarchically hyperbolic spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3CZ7WLSP}},
  note         = {Machine review of arXiv:2509.00321}
}

abstract

We prove that, under a mild assumption, any metrizable compactification of a one-ended proper geodesic metric space is connected. As a consequence, we deduce that the boundary, introduced by Durham--Hagen--Sisto, of a one-ended hierarchically hyperbolic space is connected. Moreover, we prove that the connectedness of the boundary of a hierarchically hyperbolic group is equivalent to the one-endedness of the group. As an application, we show that if, for $n\geq 2$, $G_1=A_1\ast\dots\ast A_n$ and $G_2=B_1\ast\dots\ast B_n$ are free products of one-ended hierarchically hyperbolic groups, then the boundary of $G_1$ is homeomorphic to the boundary of $G_2$ if and only if the boundary of $A_i$ is homeomorphic to the boundary of $B_i$ for $1\leq i\leq n$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 23 canonical work pages

[3]

Relative hyperbolicity, thickness, and the hierarchically hyperbolic boundary

Carolyn Abbott, Jason Behrstock, and Jacob Russell. Relative hyperbolicity, thickness, and the hierarchically hyperbolic boundary. arXiv:2305.16906, 2023. URL: https://arxiv.org/pdf/ 2305.16906

work page Pith review arXiv 2023
[7]

B. H. Bowditch. Relatively hyperbolic groups. Internat. J. Algebra and Computation. 22, 1250016, 66pp , 2012

work page 2012
[1]

Banff International Re- search Station

Advances in hierarchical hyperbolicity BIRS workshop question session. Banff International Re- search Station. Edited by C. Abbott, J. Behrstock, and J. Russell. May 2024

work page 2024
[2]

Largest acylindrical actions and stability in hierarchically hyperbolic groups

Carolyn Abbott, Jason Behrstock, and Matthew Gentry Durham. Largest acylindrical actions and stability in hierarchically hyperbolic groups. Trans. Amer. Math. Soc. Ser. B , 8:66–104,

work page
[4]

Structure invariant properties of the hi- erarchically hyperbolic boundary

Carolyn Abbott, Jason Behrstock, and Jacob Russell. Structure invariant properties of the hi- erarchically hyperbolic boundary. Journal of Topology and Analysis , pages 1–45, 2024. doi: 10.1142/S1793525323500516

doi:10.1142/s1793525323500516 2024
[5]

Hierarchically hyperbolic spaces II: Com- bination theorems and the distance formula

Jason Behrstock, Mark Hagen, and Alessandro Sisto. Hierarchically hyperbolic spaces II: Com- bination theorems and the distance formula. Pacific J. Math. , 299(2):257–338, 2019

work page 2019
[6]

Hagen, and Alessandro Sisto

Jason Behrstock, Mark F. Hagen, and Alessandro Sisto. Hierarchically hyperbolic spaces, I: Curve complexes for cubical groups. Geom. Topol., 21(3):1731–1804, 2017

work page 2017
[8]

Bridson and A Haefliger

M. Bridson and A Haefliger. Metric spaces of nonpositive curvature. Grundlehren der mathema- tischen Wissenchaften, Vol 319, Springer-Verlag , 1999

work page 1999

Show all 24 references

[9]

Homeomorphism types of the Floyd boundaries of infinite-ended groups

Subhajit Chakraborty and Ravi Tomar. Homeomorphism types of the Floyd boundaries of infinite-ended groups. arXiv:2401.00147, 2024. URL: https://arxiv.org/pdf/2401.00147

2024 arXiv
[10]

Hagen, and Alessandro Sisto

Matthew Gentry Durham, Mark F. Hagen, and Alessandro Sisto. Boundaries and automorphisms of hierarchically hyperbolic spaces. Geom. Topol., 21(6):3659–3758, 2017

2017
[11]

M. Gromov. Hyperbolic Groups. in Essays in Group Theory, ed. Gersten, MSRI Publ.,vol.8, Springer Verlag, pages 75–263, 1985

1985
[12]

Mark F. Hagen. Comment on second-countability and metrisability of the HHS boundary, 2020. URL: https://www.wescac.net/boundary_metrisable.pdf

2020
[13]

A z-structure for the mapping class group

Ursula Hamenst¨ adt. A z-structure for the mapping class group. arXiv:2505.18808, 2025

2025 arXiv
[14]

A. Martin. Non-positively curved complexes of groups and boundries. Geom. Topol. 18(1), pages 31–102, 2014

2014
[15]

Infinitely-ended hyperbolic groups with homeomor- phic Gromov boundaries

Alexandre Martin and Jacek ´Swiaãtkowski. Infinitely-ended hyperbolic groups with homeomor- phic Gromov boundaries. J. Group Theory , 18(2):273–289, 2015

2015
[16]

H. A. Masur and Y. N. Minsky. Geometry of the complex of curves. II. Hierarchical structure. Geom. Funct. Anal., 10(4):902–974, 2000. ON CONNECTEDNESS OF THE BOUNDARY OF HHS 19

2000
[17]

Masur and Yair N

Howard A. Masur and Yair N. Minsky. Geometry of the complex of curves. I. Hyperbolicity. Invent. Math. , 138(1):103–149, 1999

1999
[18]

Unbounded domains in hierarchically hyperbolic groups

Harry Petyt and Davide Spriano. Unbounded domains in hierarchically hyperbolic groups. Groups Geom. Dyn. , 17(2):479–500, 2023

2023
[19]

A combinatorial model for the Teichm¨ uller metric

Kasra Rafi. A combinatorial model for the Teichm¨ uller metric. Geom. Funct. Anal., 17(3):936– 959, 2007

2007
[20]

Convexity in hierarchically hyperbolic spaces

Jacob Russell, Davide Spriano, and Hung Cong Tran. Convexity in hierarchically hyperbolic spaces. Algebr. Geom. Topol., 23(3):1167–1248, 2023

2023
[21]

J.-P. Serre. Trees. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. Trans- lated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation

2003
[22]

Homeomorphism types of the Bowditch boundaries of infinite-ended relatively hy- perbolic groups

Ravi Tomar. Homeomorphism types of the Bowditch boundaries of infinite-ended relatively hy- perbolic groups. J. Group Theory , 28(1):51–65, 2025

2025
[23]

Morse boundaries of graphs of groups with finite edge groups

Stefanie Zbinden. Morse boundaries of graphs of groups with finite edge groups. J. Group Theory, 26(5):969–1002, 2023. Email address : ravitomar547@gmail.com Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road Haidian District, Beijin...

2023
[2021]

With an appendix by Daniel Berlyne and Jacob Russell

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [3]

Relative hyperbolicity, thickness, and the hierarchically hyperbolic boundary

Carolyn Abbott, Jason Behrstock, and Jacob Russell. Relative hyperbolicity, thickness, and the hierarchically hyperbolic boundary. arXiv:2305.16906, 2023. URL: https://arxiv.org/pdf/ 2305.16906

work page Pith review arXiv 2023

[2] [7]

B. H. Bowditch. Relatively hyperbolic groups. Internat. J. Algebra and Computation. 22, 1250016, 66pp , 2012

work page 2012

[3] [1]

Banff International Re- search Station

Advances in hierarchical hyperbolicity BIRS workshop question session. Banff International Re- search Station. Edited by C. Abbott, J. Behrstock, and J. Russell. May 2024

work page 2024

[4] [2]

Largest acylindrical actions and stability in hierarchically hyperbolic groups

Carolyn Abbott, Jason Behrstock, and Matthew Gentry Durham. Largest acylindrical actions and stability in hierarchically hyperbolic groups. Trans. Amer. Math. Soc. Ser. B , 8:66–104,

work page

[5] [4]

Structure invariant properties of the hi- erarchically hyperbolic boundary

Carolyn Abbott, Jason Behrstock, and Jacob Russell. Structure invariant properties of the hi- erarchically hyperbolic boundary. Journal of Topology and Analysis , pages 1–45, 2024. doi: 10.1142/S1793525323500516

doi:10.1142/s1793525323500516 2024

[6] [5]

Hierarchically hyperbolic spaces II: Com- bination theorems and the distance formula

Jason Behrstock, Mark Hagen, and Alessandro Sisto. Hierarchically hyperbolic spaces II: Com- bination theorems and the distance formula. Pacific J. Math. , 299(2):257–338, 2019

work page 2019

[7] [6]

Hagen, and Alessandro Sisto

Jason Behrstock, Mark F. Hagen, and Alessandro Sisto. Hierarchically hyperbolic spaces, I: Curve complexes for cubical groups. Geom. Topol., 21(3):1731–1804, 2017

work page 2017

[8] [8]

Bridson and A Haefliger

M. Bridson and A Haefliger. Metric spaces of nonpositive curvature. Grundlehren der mathema- tischen Wissenchaften, Vol 319, Springer-Verlag , 1999

work page 1999