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arxiv: 2605.10248 · v1 · submitted 2026-05-11 · 🧮 math.GR · math.MG

Recognition: 2 theorem links

· Lean Theorem

From branching quasiflats to flats in CAT(0) cube complexes

Shaked Bader , Oussama Bensaid , Harry Petyt
Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:42 UTC · model grok-4.3

classification 🧮 math.GR math.MG
keywords CAT(0) cube complexesquasiisometric embeddingsflatsTits boundaryright-angled Artin groupsbranching conditionsrigidity
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The pith

Under geometric branching conditions, quasiisometric embeddings between finite-dimensional CAT(0) cube complexes send flats to within finite Hausdorff distance of flats.

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

The paper introduces geometric branching conditions for quasiisometric embeddings of finite-dimensional CAT(0) cube complexes. These conditions ensure that flats in the domain, even those not of top rank, are mapped within finite Hausdorff distance of flats in the target. A sympathetic reader would care because the result controls the behavior of embeddings at large scales and produces maps between graphs built from the Tits boundaries. The same approach recovers known rigidity statements for symmetric spaces and Euclidean buildings of matching spherical type, and it supplies a tool for studying embeddings of right-angled Artin groups.

Core claim

We study quasiisometric embeddings between finite-dimensional CAT(0) cube complexes and introduce geometric branching conditions under which flats in the domain are mapped within finite Hausdorff distance of flats in the target, yielding embeddings between graphs associated to their Tits boundaries. The same methods recover rigidity results for quasiisometric embeddings of symmetric spaces and Euclidean buildings of the same spherical type.

What carries the argument

geometric branching conditions on the quasiisometric embedding that force lower-rank flats to stay close to flats

If this is right

  • Natural graphs associated with the Tits boundaries of the cube complexes admit embeddings induced by the original map.
  • The conditions supply a key step toward classifying quasiisometric embeddings between right-angled Artin groups.
  • Rigidity results previously known for quasiisometric embeddings of symmetric spaces are recovered by the same argument.
  • Rigidity also holds for quasiisometric embeddings of Euclidean buildings of the same spherical type.

Where Pith is reading between the lines

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

  • The branching conditions could be checked in other groups that act on CAT(0) cube complexes to detect when flats are preserved up to bounded error.
  • If the conditions turn out to hold automatically for many natural embeddings, they would imply a broader flat-rigidity theorem inside finite-dimensional cube complexes.
  • The method might extend to quasiisometries between other non-positively curved complexes where flats can be detected by branching data.

Load-bearing premise

The domain and target are finite-dimensional CAT(0) cube complexes and the quasiisometric embedding satisfies the stated geometric branching conditions.

What would settle it

A counterexample would be a quasiisometric embedding satisfying the branching conditions that sends some flat in the domain to a set remaining at infinite Hausdorff distance from every flat in the target.

Figures

Figures reproduced from arXiv: 2605.10248 by Harry Petyt, Oussama Bensaid, Shaked Bader.

Figure 1
Figure 1. Figure 1: Examples in rank 3: the red vertices define branch-complemented standard geodesics, and the red cliques define directionally branch-complemented standard flats. Another basic example is given by products of 3–regular trees. In that case, every geodesic contained in a factor is branch-complemented, and every k–flat contained in a product of k factors is directionally branch-complemented. In fact, this examp… view at source ↗
Figure 2
Figure 2. Figure 2: In the universal cover of the Salvetti complex associated with this graph, all [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A semisingular geodesic in a CAT(0) square complex with no singular geodesics. [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The simplicial complex Σ2. Lemma 8.3. Let n ∈ N, and let u and v be vertices of Σn. Let H ⊆ Σn be a subset contained in the (n − 1)–skeleton such that H ∩ {u, v} = ∅. If u and v lie in different connected components of Σn \ H, then they are antipodal. Proof. By the definition of Σn, if u and v are non-antipodal vertices, then they are neighbours and there is an n–simplex σ containing both of them. Since H … view at source ↗
Figure 5
Figure 5. Figure 5: In the right-angled Artin group given by this graph, the standard 2–flats from [PITH_FULL_IMAGE:figures/full_fig_p051_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: In the right-angled Artin group given by this graph, the standard geodesics [PITH_FULL_IMAGE:figures/full_fig_p052_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: In the right-angled Artin group associated with this graph, all standard geodesics [PITH_FULL_IMAGE:figures/full_fig_p053_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: shows the reliance between the statements in the paper leading to Theorem 10.1. In black, statements for general complete CAT(0) spaces or proofs that hold verbatim in our case. In blue, we replace Proposition 7.1 with [KL97, Prop. 7.1.1], in pink, a statement for which the proof holds for symmetric spaces of non-compact type and thick Euclidean buildings with replacing “orthants” with “Weyl cones”. In red… view at source ↗
read the original abstract

We study quasiisometric embeddings between finite-dimensional CAT(0) cube complexes. More specifically, we introduce geometric branching conditions under which flats in the domain, not necessarily of top rank, are mapped within finite Hausdorff distance of flats. As a consequence, one obtains embeddings between natural graphs associated with the Tits boundaries of those cube complexes. These results form a key step in understanding quasiisometric embeddings between right-angled Artin groups. In an appendix, we also explain how the same methods recover previously established rigidity results for quasiisometric embeddings of symmetric spaces and Euclidean buildings of the same spherical type.

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

0 major / 3 minor

Summary. The paper introduces geometric branching conditions on quasiisometric embeddings between finite-dimensional CAT(0) cube complexes under which flats in the domain (not necessarily of top rank) are mapped within finite Hausdorff distance of flats in the target. This yields embeddings of the natural graphs associated to the Tits boundaries of the complexes. The results are positioned as a key step toward understanding quasiisometric embeddings of right-angled Artin groups; an appendix recovers known rigidity statements for symmetric spaces and Euclidean buildings of the same spherical type by verifying analogous branching conditions in those settings.

Significance. If the branching conditions are correctly formulated and the implications hold, the work supplies a new, verifiable criterion for controlling quasiflats in cube complexes. This directly advances the program of quasiisometric rigidity for right-angled Artin groups and demonstrates that the same technique recovers classical results for symmetric spaces and buildings, indicating broad applicability within geometric group theory.

minor comments (3)
  1. [§2–3] The definition of the geometric branching conditions (presumably in §2 or §3) would benefit from an explicit statement of the minimal set of hyperplane-crossing and direction-set controls required; the current phrasing leaves some ambiguity about whether the conditions are local or global.
  2. [Appendix] In the appendix, the verification that symmetric spaces and buildings satisfy the branching conditions is sketched rather than fully detailed; adding a short table or diagram summarizing the checked hyperplane configurations would improve readability.
  3. [§4] Notation for the Tits boundary graphs and their embeddings is introduced without a dedicated comparison to existing literature (e.g., the boundary graphs of Caprace–Sageev or Behrstock–Hagen); a brief remark on the relation would help situate the new embeddings.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the branching conditions for controlling quasiflats, and recommendation of minor revision. No specific major comments appear in the report, so we have no individual points requiring detailed rebuttal or revision at this stage. We are pleased that the work is viewed as advancing the quasiisometric rigidity program for right-angled Artin groups while recovering classical results via the appendix.

Circularity Check

0 steps flagged

No circularity: direct construction from new branching conditions

full rationale

The paper introduces geometric branching conditions on quasiisometric embeddings between finite-dimensional CAT(0) cube complexes and proves that these force lower-rank flats to lie at finite Hausdorff distance from flats in the target, via control of hyperplane crossings and direction sets. This is a forward derivation from the stated assumptions to the claimed conclusion, with no reduction of any prediction or central result to a fitted parameter, self-definition, or load-bearing self-citation. The appendix verifies that the same conditions recover known rigidity statements for symmetric spaces and buildings, which is external validation rather than circularity. The derivation chain is self-contained against the paper's own definitions and does not rename or smuggle in prior results as new.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, invented entities, or ad-hoc axioms are stated. The work relies on standard background properties of CAT(0) cube complexes and quasiisometries.

axioms (1)
  • domain assumption Standard properties of finite-dimensional CAT(0) cube complexes and quasiisometric embeddings
    Invoked implicitly as the setting for the branching conditions and flat-mapping result.

pith-pipeline@v0.9.0 · 5401 in / 1203 out tokens · 34443 ms · 2026-05-12T04:42:20.679688+00:00 · methodology

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Works this paper leans on

300 extracted references · 300 canonical work pages

  1. [1]

    and Osin, Denis , TITLE =

    Abbott, Carolyn and Balasubramanya, Sahana H. and Osin, Denis , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2019 , NUMBER =. doi:10.2140/agt.2019.19.1747 , URL =

    work page doi:10.2140/agt.2019.19.1747 2019

  2. [2]

    Groups Geom

    Abbott, Carolyn and Behrstock, Jason , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2023 , NUMBER =. doi:10.4171/ggd/722 , URL =

    work page doi:10.4171/ggd/722 2023

  3. [3]

    , TITLE =

    Abbott, Carolyn and Behrstock, Jason and Durham, Matthew G. , TITLE =. Trans. Amer. Math. Soc. Ser. B , FJOURNAL =. 2021 , PAGES =. doi:10.1090/btran/50 , URL =

    work page doi:10.1090/btran/50 2021

  4. [4]

    Abbott, Carolyn and Behrstock, Jason and Russell, Jacob , TITLE =. J. Topol. Anal. , FJOURNAL =. 2025 , NUMBER =. doi:10.1142/S1793525323500516 , URL =

    work page doi:10.1142/s1793525323500516 2025

  5. [5]

    and Dahmani, Fran

    Abbott, Carolyn R. and Dahmani, Fran. Property. Math. Z. , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00209-018-2094-1 , URL =

    work page doi:10.1007/s00209-018-2094-1 2019

  6. [6]

    Abbott, Carolyn and Hagen, Mark and Petyt, Harry and Zalloum, Abdul , TITLE =. Anal. Geom. Metr. Spaces , FJOURNAL =. 2025 , NUMBER =. doi:10.1515/agms-2025-0027 , URL =

    work page doi:10.1515/agms-2025-0027 2025

  7. [7]

    and Manning, Jason F

    Abbott, Carolyn R. and Manning, Jason F. , TITLE =. Michigan Math. J. , FJOURNAL =. 2024 , NUMBER =. doi:10.1307/mmj/20216112 , URL =

    work page doi:10.1307/mmj/20216112 2024

  8. [8]

    and Ng, Thomas and Spriano, Davide and Gupta, Radhika and Petyt, Harry , TITLE =

    Abbott, Carolyn R. and Ng, Thomas and Spriano, Davide and Gupta, Radhika and Petyt, Harry , TITLE =. Math. Z. , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s00209-023-03411-6 , URL =

    work page doi:10.1007/s00209-023-03411-6 2024

  9. [9]

    Adyan, S. I. , TITLE =. Trudy Moskov. Mat. Ob. 1957 , PAGES =

    work page 1957

  10. [10]

    Agol, Ian , TITLE =. Doc. Math. , FJOURNAL =. 2013 , PAGES =

    work page 2013

  11. [11]

    Available at http://aimpl.org/freebycyclic , YEAR =

    AimPL , TITLE =. Available at http://aimpl.org/freebycyclic , YEAR =

    work page

  12. [12]

    Aleksandrov, A. D. , TITLE =. Dokl. Akad. Nauk SSSR , FJOURNAL =. 1970 , PAGES =

    work page 1970

  13. [13]

    Aleksandrov, A. D. and Berestovski i, V. N. and Nikolaev, I. G. , TITLE =. Uspekhi Mat. Nauk , FJOURNAL =. 1986 , NUMBER =

    work page 1986

  14. [14]

    and Bishop, Richard L

    Alexander, Stephanie B. and Bishop, Richard L. , TITLE =. Enseign. Math. (2) , FJOURNAL =. 1990 , NUMBER =

    work page 1990

  15. [15]

    Translation lengths in

    Alibegovi\'. Translation lengths in. Geom. Dedicata , FJOURNAL =. 2002 , PAGES =. doi:10.1023/A:1019695003668 , URL =

    work page doi:10.1023/a:1019695003668 2002

  16. [16]

    2011 , NUMBER =

    Strongly contracting geodesics in outer space , JOURNAL =. 2011 , NUMBER =. doi:10.2140/gt.2011.15.2181 , URL =

    work page doi:10.2140/gt.2011.15.2181 2011

  17. [17]

    Israel J

    Algom-Kfir, Yael and Hilion, Arnaud and Stark, Emily , TITLE =. Israel J. Math. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s11856-021-2191-4 , URL =

    work page doi:10.1007/s11856-021-2191-4 2021

  18. [18]

    , TITLE =

    Alonso, Juan M. , TITLE =. Algorithms and classification in combinatorial group theory (. 1992 , MRCLASS =. doi:10.1007/978-1-4613-9730-4\_7 , URL =

    work page doi:10.1007/978-1-4613-9730-4 1992

  19. [19]

    and Bridson, Martin R

    Alonso, Juan M. and Bridson, Martin R. , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 1995 , NUMBER =. doi:10.1112/plms/s3-70.1.56 , URL =

    work page doi:10.1112/plms/s3-70.1.56 1995

  20. [20]

    Actions affines isom\'

    Alvarez, Aur\'. Actions affines isom\'. Expo. Math. , FJOURNAL =. 2017 , NUMBER =. doi:10.1016/j.exmath.2016.06.005 , URL =

    work page doi:10.1016/j.exmath.2016.06.005 2017

  21. [21]

    and Aramayona, Javier and Shackleton, Kenneth J

    Anderson, James W. and Aramayona, Javier and Shackleton, Kenneth J. , TITLE =. J. Group Theory , FJOURNAL =. 2007 , NUMBER =. doi:10.1515/JGT.2007.054 , URL =

    work page doi:10.1515/jgt.2007.054 2007

  22. [22]

    Anderson, J. W. and Aramayona, J. and Shackleton, K. J. , TITLE =. In the tradition of. 2007 , MRCLASS =. doi:10.1090/conm/432/08294 , URL =

    work page doi:10.1090/conm/432/08294 2007

  23. [23]

    Andreev, P. D. , TITLE =. Sibirsk. Mat. Zh. , FJOURNAL =. 2006 , NUMBER =. doi:10.1007/s11202-006-0001-1 , URL =

    work page doi:10.1007/s11202-006-0001-1 2006

  24. [24]

    Rocky Mountain J

    Andrew, Naomi and Hughes, Sam and Kudlinska, Monika , TITLE =. Rocky Mountain J. Math. , FJOURNAL =. 2024 , NUMBER =. doi:10.1216/rmj.2024.54.933 , URL =

    work page doi:10.1216/rmj.2024.54.933 2024

  25. [25]

    Andrew, Naomi and Martino, Armando , TITLE =. J. Algebra , FJOURNAL =. 2022 , PAGES =. doi:10.1016/j.jalgebra.2022.03.033 , URL =

    work page doi:10.1016/j.jalgebra.2022.03.033 2022

  26. [26]

    Commensurating endomorphisms of acylindrically hyperbolic groups and applications , JOURNAL =

    Antol\'. Commensurating endomorphisms of acylindrically hyperbolic groups and applications , JOURNAL =. 2016 , NUMBER =. doi:10.4171/GGD/379 , URL =

    work page doi:10.4171/ggd/379 2016

  27. [27]

    Intersection properties of stable subgroups and bounded cohomology , JOURNAL =

    Antol\'. Intersection properties of stable subgroups and bounded cohomology , JOURNAL =. 2019 , NUMBER =. doi:10.1512/iumj.2019.68.7592 , URL =

    work page doi:10.1512/iumj.2019.68.7592 2019

  28. [28]

    Aougab, Tarik , TITLE =. Geom. Topol. , FJOURNAL =. 2013 , NUMBER =. doi:10.2140/gt.2013.17.2855 , URL =

    work page doi:10.2140/gt.2013.17.2855 2013

  29. [29]

    and Taylor, Samuel J

    Aougab, Tarik and Durham, Matthew G. and Taylor, Samuel J. , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2017 , NUMBER =. doi:10.1112/jlms.12071 , URL =

    work page doi:10.1112/jlms.12071 2017

  30. [30]

    , TITLE =

    Aougab, Tarik and Taylor, Samuel J. , TITLE =. In the tradition of. 2017 , MRCLASS =. doi:10.1090/conm/696/14014 , URL =

    work page doi:10.1090/conm/696/14014 2017

  31. [31]

    and Souto, Juan , TITLE =

    Aramayona, Javier and Leininger, Christopher J. and Souto, Juan , TITLE =. Geom. Topol. , FJOURNAL =. 2009 , NUMBER =. doi:10.2140/gt.2009.13.2523 , URL =

    work page doi:10.2140/gt.2009.13.2523 2009

  32. [32]

    Michigan Math

    Aramayona, Javier and Souto, Juan , TITLE =. Michigan Math. J. , FJOURNAL =. 2011 , NUMBER =. doi:10.1307/mmj/1320763044 , URL =

    work page doi:10.1307/mmj/1320763044 2011

  33. [33]

    Aramayona, Javier and Souto, Juan , TITLE =. Geom. Topol. , FJOURNAL =. 2012 , NUMBER =. doi:10.2140/gt.2012.16.2285 , URL =

    work page doi:10.2140/gt.2012.16.2285 2012

  34. [34]

    Ardila, Federico and Owen, Megan and Sullivant, Seth , TITLE =. Adv. in Appl. Math. , FJOURNAL =. 2012 , NUMBER =. doi:10.1016/j.aam.2011.06.004 , URL =

    work page doi:10.1016/j.aam.2011.06.004 2012

  35. [35]

    and Panitchpakdi, P

    Aronszajn, N. and Panitchpakdi, P. , TITLE =. Pacific J. Math. , FJOURNAL =. 1956 , PAGES =

    work page 1956

  36. [36]

    and Cashen, Christopher H

    Arzhantseva, Goulnara N. and Cashen, Christopher H. and Gruber, Dominik and Hume, David , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2019 , NUMBER =. doi:10.4171/GGD/498 , URL =

    work page doi:10.4171/ggd/498 2019

  37. [37]

    and Cashen, Christopher H

    Arzhantseva, Goulnara N. and Cashen, Christopher H. and Tao, Jing , TITLE =. Pacific J. Math. , FJOURNAL =. 2015 , NUMBER =. doi:10.2140/pjm.2015.278.1 , URL =

    work page doi:10.2140/pjm.2015.278.1 2015

  38. [38]

    and Wright, Nick and Zhang, Jiawen , TITLE =

    Arzhantseva, Goulnara and Niblo, Graham A. and Wright, Nick and Zhang, Jiawen , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2018 , NUMBER =. doi:10.2140/agt.2018.18.493 , URL =

    work page doi:10.2140/agt.2018.18.493 2018

  39. [39]

    Avann, S. P. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1961 , PAGES =. doi:10.2307/2034206 , URL =

    work page doi:10.2307/2034206 1961

  40. [40]

    , TITLE =

    Baez, John C. , TITLE =. Bull. Amer. Math. Soc. (N.S.) , FJOURNAL =. 2002 , NUMBER =. doi:10.1090/S0273-0979-01-00934-X , URL =

    work page doi:10.1090/s0273-0979-01-00934-x 2002

  41. [41]

    Baik, Hyungryul and Seo, Donggyun and Shin, Hyunshik , TITLE =. Geom. Dedicata , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s10711-022-00736-0 , URL =

    work page doi:10.1007/s10711-022-00736-0 2023

  42. [42]

    , TITLE =

    Balasubramanya, Sahana H. , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.2140/agt.2017.17.2145 , URL =

    work page doi:10.2140/agt.2017.17.2145 2017

  43. [43]

    Balasubramanya, Sahana and Chesser, Marissa and Kerr, Alice and Mangahas, Johanna and Trin, Marie , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2025 , NUMBER =. doi:10.1090/proc/17280 , URL =

    work page doi:10.1090/proc/17280 2025

  44. [44]

    and Fournier-Facio, Francesco and Genevois, Anthony , TITLE =

    Balasubramanya, Sahana H. and Fournier-Facio, Francesco and Genevois, Anthony , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2025 , NUMBER =. doi:10.4171/ggd/806 , URL =

    work page doi:10.4171/ggd/806 2025

  45. [45]

    Sharp uniform convexity and smoothness inequalities for trace norms,

    Ball, Keith and Carlen, Eric A. and Lieb, Elliott H. , TITLE =. Invent. Math. , FJOURNAL =. 1994 , NUMBER =. doi:10.1007/BF01231769 , URL =

    work page doi:10.1007/bf01231769 1994

  46. [46]

    1995 , PAGES =

    Ballmann, Werner , TITLE =. 1995 , PAGES =. doi:10.1007/978-3-0348-9240-7 , URL =

    work page doi:10.1007/978-3-0348-9240-7 1995

  47. [47]

    Ballmann, Werner , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1985 , NUMBER =. doi:10.2307/1971331 , URL =

    work page doi:10.2307/1971331 1985

  48. [48]

    Ballmann, Werner and Brin, Misha and Eberlein, Patrick , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1985 , NUMBER =. doi:10.2307/1971373 , URL =

    work page doi:10.2307/1971373 1985

  49. [49]

    Ballmann, Werner and Brin, Misha and Spatzier, Ralf , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1985 , NUMBER =. doi:10.2307/1971303 , URL =

    work page doi:10.2307/1971303 1985

  50. [50]

    Geometric and probabilistic structures in dynamics , SERIES =

    Ballmann, Werner and Buyalo, Sergei , TITLE =. Geometric and probabilistic structures in dynamics , SERIES =. 2008 , MRCLASS =. doi:10.1090/conm/469/09159 , URL =

    work page doi:10.1090/conm/469/09159 2008

  51. [51]

    1985 , PAGES =

    Ballmann, Werner and Gromov, Mikhael and Schroeder, Viktor , TITLE =. 1985 , PAGES =. doi:10.1007/978-1-4684-9159-3 , URL =

    work page doi:10.1007/978-1-4684-9159-3 1985

  52. [52]

    Surveys on discrete and computational geometry , SERIES =

    Bandelt, Hans-J\"urgen and Chepoi, Victor , TITLE =. Surveys on discrete and computational geometry , SERIES =. 2008 , ISBN =. doi:10.1090/conm/453/08795 , URL =

    work page doi:10.1090/conm/453/08795 2008

  53. [53]

    and Hedl\'

    Bandelt, Hans-J. and Hedl\'. Median algebras , JOURNAL =. 1983 , NUMBER =. doi:10.1016/0012-365X(83)90173-5 , URL =

    work page doi:10.1016/0012-365x(83)90173-5 1983

  54. [54]

    and Vel, M

    Bandelt, H.-J. and Vel, M. van de , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 1991 , NUMBER =. doi:10.1016/0097-3165(91)90044-H , URL =

    work page doi:10.1016/0097-3165(91)90044-h 1991

  55. [55]

    Bartels, Arthur and Bestvina, Mladen , TITLE =. Invent. Math. , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00222-018-0834-9 , URL =

    work page doi:10.1007/s00222-018-0834-9 2019

  56. [56]

    , TITLE =

    Bass, H. , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 1972 , PAGES =. doi:10.1112/plms/s3-25.4.603 , URL =

    work page doi:10.1112/plms/s3-25.4.603 1972

  57. [57]

    Basso, Giuliano , TITLE =. Enseign. Math. , FJOURNAL =. 2024 , NUMBER =. doi:10.4171/lem/1043 , URL =

    work page doi:10.4171/lem/1043 2024

  58. [58]

    and Gersten, S

    Baumslag, G. and Gersten, S. M. and Shapiro, M. and Short, H. , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 1991 , NUMBER =. doi:10.1016/0022-4049(91)90139-S , URL =

    work page doi:10.1016/0022-4049(91)90139-s 1991

  59. [59]

    Beckman, F. S. and Quarles, Jr., D. A. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1953 , PAGES =. doi:10.2307/2032415 , URL =

    work page doi:10.2307/2032415 1953

  60. [60]

    Behrstock, Jason , TITLE =. Geom. Topol. , FJOURNAL =. 2006 , PAGES =. doi:10.2140/gt.2006.10.1523 , URL =

    work page doi:10.2140/gt.2006.10.1523 2006

  61. [61]

    Behrstock, Jason and Charney, Ruth , TITLE =. Math. Ann. , FJOURNAL =. 2012 , NUMBER =. doi:10.1007/s00208-011-0641-8 , URL =

    work page doi:10.1007/s00208-011-0641-8 2012

  62. [62]

    Combinatorial higher dimensional isoperimetry and divergence , JOURNAL =

    Behrstock, Jason and Dru. Combinatorial higher dimensional isoperimetry and divergence , JOURNAL =. 2019 , NUMBER =. doi:10.1142/S1793525319500225 , URL =

    work page doi:10.1142/s1793525319500225 2019

  63. [63]

    Divergence, thick groups, and short conjugators , JOURNAL =

    Behrstock, Jason and Dru. Divergence, thick groups, and short conjugators , JOURNAL =. 2014 , NUMBER =

    work page 2014

  64. [64]

    Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity , JOURNAL =

    Behrstock, Jason and Dru. Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity , JOURNAL =. 2009 , NUMBER =. doi:10.1007/s00208-008-0317-1 , URL =

    work page doi:10.1007/s00208-008-0317-1 2009

  65. [65]

    Median structures on asymptotic cones and homomorphisms into mapping class groups , JOURNAL =

    Behrstock, Jason and Dru. Median structures on asymptotic cones and homomorphisms into mapping class groups , JOURNAL =. 2011 , NUMBER =. doi:10.1112/plms/pdq025 , URL =

    work page doi:10.1112/plms/pdq025 2011

  66. [66]

    Behrstock, Jason and Hagen, Mark and Martin, Alexandre and Sisto, Alessandro , TITLE =. J. Topol. , FJOURNAL =. 2024 , NUMBER =. doi:10.1112/topo.12351 , URL =

    work page doi:10.1112/topo.12351 2024

  67. [67]

    Behrstock, Jason and Hagen, Mark and Sisto, Alessandro , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2017 , NUMBER =. doi:10.1112/plms.12026 , URL =

    work page doi:10.1112/plms.12026 2017

  68. [68]

    Behrstock, Jason and Hagen, Mark and Sisto, Alessandro , TITLE =. Geom. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.2140/gt.2017.21.1731 , URL =

    work page doi:10.2140/gt.2017.21.1731 2017

  69. [69]

    Pacific J

    Behrstock, Jason and Hagen, Mark and Sisto, Alessandro , TITLE =. Pacific J. Math. , FJOURNAL =. 2019 , NUMBER =. doi:10.2140/pjm.2019.299.257 , URL =

    work page doi:10.2140/pjm.2019.299.257 2019

  70. [70]

    Duke Math

    Behrstock, Jason and Hagen, Mark and Sisto, Alessandro , TITLE =. Duke Math. J. , FJOURNAL =. 2021 , NUMBER =. doi:10.1215/00127094-2020-0056 , URL =

    work page doi:10.1215/00127094-2020-0056 2021

  71. [71]

    and Mosher, Lee , TITLE =

    Behrstock, Jason and Kleiner, Bruce and Minsky, Yair N. and Mosher, Lee , TITLE =. Geom. Topol. , FJOURNAL =. 2012 , NUMBER =. doi:10.2140/gt.2012.16.781 , URL =

    work page doi:10.2140/gt.2012.16.781 2012

  72. [72]

    , TITLE =

    Behrstock, Jason and Minsky, Yair N. , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2011 , NUMBER =. doi:10.1112/jlms/jdr027 , URL =

    work page doi:10.1112/jlms/jdr027 2011

  73. [73]

    and Minsky, Yair N

    Behrstock, Jason A. and Minsky, Yair N. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2008 , NUMBER =. doi:10.4007/annals.2008.167.1055 , URL =

    work page doi:10.4007/annals.2008.167.1055 2008

  74. [74]

    and Januszkiewicz, Tadeusz and Neumann, Walter D

    Behrstock, Jason A. and Januszkiewicz, Tadeusz and Neumann, Walter D. , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2010 , NUMBER =. doi:10.4171/GGD/100 , URL =

    work page doi:10.4171/ggd/100 2010

  75. [75]

    and Neumann, Walter D

    Behrstock, Jason A. and Neumann, Walter D. , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2012 , PAGES =. doi:10.1515/crelle.2011.143 , URL =

    work page doi:10.1515/crelle.2011.143 2012

  76. [76]

    2008 , PAGES =

    Bekka, Bachir and Harpe, Pierre de la and Valette, Alain , TITLE =. 2008 , PAGES =. doi:10.1017/CBO9780511542749 , URL =

    work page doi:10.1017/cbo9780511542749 2008

  77. [77]

    and Dranishnikov, A

    Bell, G. and Dranishnikov, A. , TITLE =. Topology Appl. , FJOURNAL =. 2008 , NUMBER =. doi:10.1016/j.topol.2008.02.011 , URL =

    work page doi:10.1016/j.topol.2008.02.011 2008

  78. [78]

    and Dranishnikov, A

    Bell, G. and Dranishnikov, A. , TITLE =. Geom. Dedicata , FJOURNAL =. 2004 , PAGES =. doi:10.1023/B:GEOM.0000013843.53884.77 , URL =

    work page doi:10.1023/b:geom.0000013843.53884.77 2004

  79. [79]

    and Fujiwara, Koji , TITLE =

    Bell, Gregory C. and Fujiwara, Koji , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2008 , NUMBER =. doi:10.1112/jlms/jdm090 , URL =

    work page doi:10.1112/jlms/jdm090 2008

  80. [80]

    Benjamini, Itai and Shamov, Alexander , TITLE =. Anal. Geom. Metr. Spaces , FJOURNAL =. 2015 , NUMBER =. doi:10.1515/agms-2015-0018 , URL =

    work page doi:10.1515/agms-2015-0018 2015

Showing first 80 references.