Download this article
 Download this article For screen
For printing
Recent Issues

Volume 30
Issue 4, 1203–1610
Issue 3, 835–1201
Issue 2, 389–833
Issue 1, 1–388

Volume 29, 9 issues

Volume 28, 9 issues

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
 
Author index
To appear
 
Other MSP journals
Integrable measure equivalence rigidity of right-angled Artin groups via quasi-isometry

Camille Horbez and Jingyin Huang

Geometry & Topology 30 (2026) 1451–1514
DOI: 10.2140/gt.2026.30.1451
Bibliography
1 P Abramenko, K S Brown, Buildings: theory and applications, 248, Springer (2008) MR2439729
2 I Agol, The virtual Haken conjecture, Doc. Math. 18 (2013) 1045 MR3104553
3 A Arhangel’skii, M Tkachenko, Topological groups and related structures, 1, Atlantis (2008) MR2433295
4 T Austin, Behaviour of entropy under bounded and integrable orbit equivalence, Geom. Funct. Anal. 26 (2016) 1483 MR3579704
5 T Austin, Integrable measure equivalence for groups of polynomial growth, Groups Geom. Dyn. 10 (2016) 117 MR3460333
6 U Bader, A Furman, R Sauer, Integrable measure equivalence and rigidity of hyperbolic lattices, Invent. Math. 194 (2013) 313 MR3117525
7 U Bader, P E Caprace, T Gelander, S Mozes, Lattices in amenable groups, Fund. Math. 246 (2019) 217 MR3959251
8 U Bader, A Furman, R Sauer, Lattice envelopes, Duke Math. J. 169 (2020) 213 MR4057144
9 J Behrstock, R Charney, Divergence and quasimorphisms of right-angled Artin groups, Math. Ann. 352 (2012) 339 MR2874959
10 J A Behrstock, W D Neumann, Quasi-isometric classification of graph manifold groups, Duke Math. J. 141 (2008) 217 MR2376814
11 J A Behrstock, T Januszkiewicz, W D Neumann, Quasi-isometric classification of some high dimensional right-angled Artin groups, Groups Geom. Dyn. 4 (2010) 681 MR2727658
12 J Behrstock, B Kleiner, Y Minsky, L Mosher, Geometry and rigidity of mapping class groups, Geom. Topol. 16 (2012) 781 MR2928983
13 R M Belinskaja, Partitionings of a Lebesgue space into trajectories which may be defined by ergodic automorphisms, Funkcional. Anal. i Priložen. 2 (1968) 4 MR245756
14 M Bestvina, B Kleiner, M Sageev, The asymptotic geometry of right-angled Artin groups, I, Geom. Topol. 12 (2008) 1653 MR2421136
15 L Bowen, L1-measure equivalence and group growth, (2016) MR3460333
16 M R Bridson, A Haefliger, Metric spaces of non-positive curvature, 319, Springer (1999) MR1744486
17 J Brodzki, S J Campbell, E Guentner, G A Niblo, N J Wright, Property A and CAT(0) cube complexes, J. Funct. Anal. 256 (2009) 1408 MR2490224
18 K S Brown, Presentations for groups acting on simply-connected complexes, J. Pure Appl. Algebra 32 (1984) 1 MR739633
19 M Burger, S Mozes, Groups acting on trees: from local to global structure, Inst. Hautes Études Sci. Publ. Math. (2000) 113 MR1839488
20 M Burger, S Mozes, Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. (2000) 151 MR1839489
21 P E Caprace, T De Medts, Lattice envelopes of right-angled Artin groups, (2024) arXiv:2401.15943
22 P E Caprace, D Hume, Orthogonal forms of Kac–Moody groups are acylindrically hyperbolic, Ann. Inst. Fourier (Grenoble) 65 (2015) 2613 MR3449592
23 P E Caprace, A Lytchak, At infinity of finite-dimensional CAT(0) spaces, Math. Ann. 346 (2010) 1 MR2558883
24 P E Caprace, M Sageev, Rank rigidity for CAT(0) cube complexes, Geom. Funct. Anal. 21 (2011) 851 MR2827012
25 R Charney, An introduction to right-angled Artin groups, Geom. Dedicata 125 (2007) 141 MR2322545
26 R Charney, M W Davis, Finite K(π,1)s for Artin groups, from: "Prospects in topology" (editor F Quinn), Ann. of Math. Stud. 138, Princeton Univ. Press (1995) 110 MR1368655
27 R Charney, M Farber, Random groups arising as graph products, Algebr. Geom. Topol. 12 (2012) 979 MR2928902
28 R Charney, J Crisp, K Vogtmann, Automorphisms of 2-dimensional right-angled Artin groups, Geom. Topol. 11 (2007) 2227 MR2372847
29 Y Cornulier, Commability and focal locally compact groups, Indiana Univ. Math. J. 64 (2015) 115 MR3320521
30 Y d Cornulier, On the quasi-isometric classification of locally compact groups, from: "New directions in locally compact groups" (editors P E Caprace, N Monod), London Math. Soc. Lecture Note Ser. 447, Cambridge Univ. Press (2018) 275 MR3793294
31 M W Davis, Buildings are CAT(0), from: "Geometry and cohomology in group theory" (editors P H Kropholler, G A Niblo, R Stöhr), London Math. Soc. Lecture Note Ser. 252, Cambridge Univ. Press (1998) 108 MR1709955
32 M B Day, Finiteness of outer automorphism groups of random right-angled Artin groups, Algebr. Geom. Topol. 12 (2012) 1553 MR2966695
33 T Delabie, J Koivisto, F Le Maître, R Tessera, Quantitative measure equivalence between amenable groups, Ann. H. Lebesgue 5 (2022) 1417 MR4526258
34 H A Dye, On groups of measure preserving transformations, I, Amer. J. Math. 81 (1959) 119 MR131516
35 H A Dye, On groups of measure preserving transformations, II, Amer. J. Math. 85 (1963) 551 MR158048
36 R Engelking, General topology, 6, Heldermann (1989) MR1039321
37 A Escalier, C Horbez, Graph products and measure equivalence: classification, rigidity, and quantitative aspects, (2024) arXiv:2401.04635
38 A Eskin, Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces, J. Amer. Math. Soc. 11 (1998) 321 MR1475886
39 A Eskin, B Farb, Quasi-flats and rigidity in higher rank symmetric spaces, J. Amer. Math. Soc. 10 (1997) 653 MR1434399
40 B Farb, C Hruska, A Thomas, Problems on automorphism groups of nonpositively curved polyhedral complexes and their lattices, from: "Geometry, rigidity, and group actions" (editors B Farb, D Fisher), Univ. Chicago Press (2011) 515 MR2807842
41 T Fernós, The Furstenberg–Poisson boundary and CAT(0) cube complexes, Ergodic Theory Dynam. Systems 38 (2018) 2180 MR3833346
42 T Fernós, J Lécureux, F Mathéus, Random walks and boundaries of CAT(0) cubical complexes, Comment. Math. Helv. 93 (2018) 291 MR3811753
43 A Furman, Gromov’s measure equivalence and rigidity of higher rank lattices, Ann. of Math. (2) 150 (1999) 1059 MR1740986
44 A Furman, Orbit equivalence rigidity, Ann. of Math. (2) 150 (1999) 1083 MR1740985
45 A Furman, Mostow–Margulis rigidity with locally compact targets, Geom. Funct. Anal. 11 (2001) 30 MR1829641
46 H Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963) 335 MR146298
47 D Gaboriau, Invariants l2 de relations d’équivalence et de groupes, Publ. Math. Inst. Hautes Études Sci. (2002) 93 MR1953191
48 E Godelle, Parabolic subgroups of Artin groups of type FC, Pacific J. Math. 208 (2003) 243 MR1971664
49 E R Green, Graph products of groups, PhD thesis, University of Leeds (1990)
50 M Gromov, Asymptotic invariants of infinite groups, from: "Geometric group theory, II" (editors G A Niblo, M A Roller), London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1 MR1253544
51 V Guirardel, C Horbez, Measure equivalence rigidity of Out(FN), (2021) arXiv:2103.03696
52 F Haglund, D T Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008) 1551 MR2377497
53 F Haglund, D T Wise, A combination theorem for special cube complexes, Ann. of Math. (2) 176 (2012) 1427 MR2979855
54 U Hamenstaedt, Geometry of the mapping class groups, III: Quasi-isometric rigidity, (2005) arXiv:math/0512429
55 C Horbez, J Huang, Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type, (2020) arXiv:2004.09325
56 C Horbez, J Huang, Measure equivalence classification of transvection-free right-angled Artin groups, J. Éc. polytech. Math. 9 (2022) 1021 MR4443237
57 C Horbez, J Huang, Measure equivalence rigidity among the Higman groups, (2022) arXiv:2206.00884
58 C Horbez, J Huang, A Ioana, Orbit equivalence rigidity of irreducible actions of right-angled Artin groups, Compos. Math. 159 (2023) 860 MR4571575
59 J Huang, Quasi-isometry classification of right-angled Artin groups, II: Several infinite out cases, (2016) arXiv:1603.02372
60 J Huang, Quasi-isometric classification of right-angled Artin groups, I : The finite out case, Geom. Topol. 21 (2017) 3467 MR3692971
61 J Huang, Commensurability of groups quasi-isometric to RAAGs, Invent. Math. 213 (2018) 1179 MR3842063
62 J Huang, B Kleiner, Groups quasi-isometric to right-angled Artin groups, Duke Math. J. 167 (2018) 537 MR3761106
63 J Huang, M Mj, Indiscrete common commensurators, (2023) arXiv:2310.04876
64 S Hughes, Graphs and complexes of lattices, (2021) arXiv:2104.13728
65 N V Ivanov, Automorphism of complexes of curves and of Teichmüller spaces, Internat. Math. Res. Notices (1997) 651 MR1460387
66 J Justin, Groupes et semi-groupes à croissance linéaire, C. R. Acad. Sci. Paris Sér. A-B 273 (1971) MR289689
67 A Kar, M Sageev, Ping pong on CAT(0) cube complexes, Comment. Math. Helv. 91 (2016) 543 MR3541720
68 A S Kechris, Classical descriptive set theory, 156, Springer (1995) MR1321597
69 D Kerr, H Li, Entropy, Shannon orbit equivalence, and sparse connectivity, Math. Ann. 380 (2021) 1497 MR4297192
70 D Kerr, H Li, Entropy, virtual Abelianness and Shannon orbit equivalence, Ergodic Theory Dynam. Systems 44 (2024) 3481 MR4818933
71 Y Kida, Introduction to measurable rigidity of mapping class groups, from: "Handbook of Teichmüller theory, II" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc. (2009) 297 MR2497783
72 Y Kida, Measure equivalence rigidity of the mapping class group, Ann. of Math. (2) 171 (2010) 1851 MR2680399
73 Y Kida, Rigidity of amalgamated free products in measure equivalence, J. Topol. 4 (2011) 687 MR2832574
74 S h Kim, T Koberda, Embedability between right-angled Artin groups, Geom. Topol. 17 (2013) 493 MR3039768
75 B Kleiner, B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. (1997) 115 MR1608566
76 M R Laurence, A generating set for the automorphism group of a graph group, J. London Math. Soc. (2) 52 (1995) 318 MR1356145
77 N Lazarovich, I Levcovitz, A Margolis, Counting lattices in products of trees, Comment. Math. Helv. 98 (2023) 597 MR4668544
78 A Margolis, Quasi-isometry classification of right-angled Artin groups that split over cyclic subgroups, Groups Geom. Dyn. 14 (2020) 1351 MR4186478
79 G A Margulis, Discrete subgroups of semisimple Lie groups, 17, Springer (1991) MR1090825
80 N Monod, Y Shalom, Orbit equivalence rigidity and bounded cohomology, Ann. of Math. (2) 164 (2006) 825 MR2259246
81 D S Ornstein, B Weiss, Ergodic theory of amenable group actions, I : The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980) 161 MR551753
82 N Radu, New simple lattices in products of trees and their projections, Canad. J. Math. 72 (2020) 1624 MR4176704
83 M S Raghunathan, Discrete subgroups of Lie groups, 68, Springer (1972) MR507234
84 M A Roller, Poc sets, median algebras and group actions : an extended study of Dunwoody’s construction and Sageev’s theorem, Habilitationsschrift, Universität Regensburg (1998) arXiv:1607.07747
85 M Sageev, CAT(0) cube complexes and groups, from: "Geometric group theory" (editors M Bestvina, M Sageev, K Vogtmann), IAS/Park City Math. Ser. 21, Amer. Math. Soc. (2014) 7 MR3329724
86 M Salvetti, Topology of the complement of real hyperplanes in CN, Invent. Math. 88 (1987) 603 MR884802
87 H Servatius, Automorphisms of graph groups, J. Algebra 126 (1989) 34 MR1023285
88 Y Shalom, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000) 1 MR1767270
89 Y Shalom, Harmonic analysis, cohomology, and the large-scale geometry of amenable groups, Acta Math. 192 (2004) 119 MR2096453
90 S Shepherd, Commensurability of lattices in right-angled buildings, Adv. Math. 441 (2024) 109522 MR4708144
91 C T C Wall, Poincaré complexes, I, Ann. of Math. (2) 86 (1967) 213 MR217791
92 R J Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups, Ann. of Math. (2) 112 (1980) 511 MR595205
93 R J Zimmer, Ergodic theory and semisimple groups, 81, Birkhäuser (1984) MR776417