Geodesic
Rigidity phenomena and the statistical properties of group actions on $\text {CAT}(0)$ cube complexes
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e150

Voir la notice de l'article provenant de la source Cambridge University Press

We compare the marked length spectra of some pairs of proper and cocompact cubical actions of a nonvirtually cyclic group on $\mathrm {CAT}(0)$ cube complexes. The cubulations are required to be virtually co-special, have the same sets of convex-cocompact subgroups, and admit a contracting element. There are many groups for which these conditions are always fulfilled for any pair of cubulations, including nonelementary cubulable hyperbolic groups, many cubulable relatively hyperbolic groups, and many right-angled Artin and Coxeter groups.For these pairs of cubulations, we study the Manhattan curve associated to their combinatorial metrics. We prove that this curve is analytic and convex, and a straight line if and only if the marked length spectra are homothetic. The same result holds if we consider invariant combinatorial metrics in which the lengths of the edges are not necessarily one. In addition, for their standard combinatorial metrics, we prove a large deviations theorem with shrinking intervals for their marked length spectra. We deduce the same result for pairs of word metrics on hyperbolic groups.The main tool is the construction of a finite-state automaton that simultaneously encodes the marked length spectra of both cubulations in a coherent way, in analogy with results about (bi)combable functions on hyperbolic groups by Calegari-Fujiwara [14]. The existence of this automaton allows us to apply the machinery of thermodynamic formalism for suspension flows over subshifts of finite type, from which we deduce our results. 
Cantrell, Stephen; Reyes, Eduardo. Rigidity phenomena and the statistical properties of group actions on $\text {CAT}(0)$ cube complexes. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e150. doi: 10.1017/fms.2025.10094
@article{10_1017_fms_2025_10094,
     author = {Cantrell, Stephen and Reyes, Eduardo},
     title = {Rigidity phenomena and the statistical properties of group actions on $\text {CAT}(0)$ cube complexes},
     journal = {Forum of Mathematics, Sigma},
     pages = {e150},
     year = {2025},
     volume = {13},
     number = {1},
     doi = {10.1017/fms.2025.10094},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10094/}
}
TY  - JOUR
AU  - Cantrell, Stephen
AU  - Reyes, Eduardo
TI  - Rigidity phenomena and the statistical properties of group actions on $\text {CAT}(0)$ cube complexes
JO  - Forum of Mathematics, Sigma
PY  - 2025
SP  - e150
VL  - 13
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10094/
DO  - 10.1017/fms.2025.10094
ID  - 10_1017_fms_2025_10094
ER  - 
%0 Journal Article
%A Cantrell, Stephen
%A Reyes, Eduardo
%T Rigidity phenomena and the statistical properties of group actions on $\text {CAT}(0)$ cube complexes
%J Forum of Mathematics, Sigma
%D 2025
%P e150
%V 13
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10094/
%R 10.1017/fms.2025.10094
%F 10_1017_fms_2025_10094

[1] Agol, I., ‘The virtual Haken conjecture’. With an appendix by I. Agol, D. Groves and J. F. Manning. Doc. Math. 18 (2013), 1045–1087.10.4171/dm/421 Google Scholar | DOI

[2] Aougab, T., Clay, M. and Rieck, Y., ‘Thermodynamic metrics on outer space’, Ergodic Theory Dynam. Systems 43(3) (2023), 729–793. Google Scholar | DOI

[3] Bergeron, N. and Wise, D. T., ‘A boundary criterion for cubulation’, Amer. J. Math. 134(3) (2012), 843–859. Google Scholar | DOI

[4] Beyrer, J. and Fioravanti, E., ‘Cross ratios and cubulations of hyperbolic groups’, Math. Ann. 384(3–4) (2022), 1547–1592. Google Scholar | DOI

[5] Beyrer, J. and Fioravanti, E., ‘Cross-ratios on CAT(0) cube complexes and marked length-spectrum rigidity’, J. Lond. Math. Soc. (2) 104(5) (2021), 1973–2015. Google Scholar | DOI

[6] Bray, H., Canary, R. and Kao, L.-Y., ‘Pressure metrics for deformation spaces of quasifuchsian groups with parabolics’, Algebr. Geom. Topol. 23(8) (2023), 3615–3653.10.2140/agt.2023.23.3615 Google Scholar | DOI

[7] Bray, H., Canary, R., Kao, L.-Y. and Martone, G., ‘Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations’, J. Reine Angew. Math. 791 (2022), 1–51.10.1515/crelle-2022-0035 Google Scholar | DOI

[8] Bregman, C., Charney, R. and Vogtmann, K., ‘Outer space for RAAGs’, Duke Math. J. 172(6) (2023), 1033–1108.10.1215/00127094-2023-0007 Google Scholar | DOI

[9] Bridgeman, M., Canary, R., Labourie, F. and Sambarino, A., ‘The pressure metric for Anosov representations’, Geom. Funct. Anal. 25(4) (2015), 1089–1179. Google Scholar | DOI

[10] Bridson, M. and Haefliger, A., Metric spaces of non-positive curvature . Grundlehren der Mathematischen Wissenschaften, vol. 319 (Springer-Verlag, Berlin, 1999). Google Scholar | DOI

[11] Burago, D., Burago, Y. and Ivanov, S., A Course in Metric Geometry . Graduate Studies in Mathematics, vol. 33 (American Mathematical Society, Providence, RI, 2001).10.1090/gsm/033 Google Scholar | DOI

[12] Burger, M., ‘Intersection, the Manhattan curve, and Patterson-Sullivan theory in rank 2’, Int. Math. Res. Not. 1993(7), 217–225. Google Scholar | DOI

[13] Bowen, R. and Series, C., ‘Markov maps associated with Fuchsian groups’, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 153–170. Google Scholar | DOI

[14] Calegari, D. and Fujiwara, K., ‘Combable functions, quasimorphisms, and the central limit theorem’, Ergodic Theory Dynam. Systems 30(5) (2010), 1343–1369. Google Scholar | DOI

[15] Cannon, J. W., ‘The combinatorial structure of cocompact discrete hyperbolic groups’, Geom. Dedicata 16(2) (1984), 123–148. Google Scholar | DOI

[16] Cantrell, S., ‘Mixing of the Mineyev flow, orbital counting and Poincaré series for strongly hyperbolic metrics’, Math. Ann. 392 (2025), 1253–1288.10.1007/s00208-025-03127-4 Google Scholar | DOI

[17] Cantrell, S. and Reyes, E., Marked length spectrum rigidity from rigidity on subsets. , 2023. Google Scholar | arXiv

[18] S. Cantrell And, R, ‘Tanaka, Invariant measures of the topological flow and measures at infinity on hyperbolic groups’, J. Mod. Dyn. 20 (2024), 215–274. Google Scholar | DOI

[19] Cantrell, S. and Tanaka, R., ‘The Manhattan curve, ergodic theory of topological flows and rigidity’, To appear in Geom. Topol., , 2021. Google Scholar | arXiv

[20] Caprace, P.-E. and Sageev, M., ‘Rank rigidity for CAT(0) cube complexes’, Geom. Funct. Anal. 21 (2011), 851–891. Google Scholar

[21] Charney, R. and Davis, M. W., ‘The K(π,1)-problem for hyperplane complements associated to infinite reflection groups’, J. Amer. Math. Soc. 8(3) (1995), 597–627. Google Scholar

[22] Charney, R., Stambaugh, N. and Vogtmann, K., ‘Outer space for untwisted automorphisms of right-angled Artin groups’, Geom. Topol. 21 (2017), 1131–1178. Google Scholar

[23] Chepoi, V., ‘Graphs of some CAT(0) complexes’, Adv. Appl. Math. 24(2) (2000), 125–179. Google Scholar | DOI

[24] Cooper, D. and Futer, D., ‘Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3–manifolds’, Geom. Topol. 23(1) (2019), 241–298.10.2140/gt.2019.23.241 Google Scholar | DOI

[25] Culler, M. and Vogtmann, K., ‘Moduli of graphs and automorphisms of free groups’, Invent. Math. 84(1) (1986), 91–119.10.1007/BF01388734 Google Scholar | DOI

[26] Dahmani, F., Futer, D. and Wise, D. T., ‘Growth of quasiconvex subgroups’, Math. Proc. Cambridge Philos. Soc. 167(3) (2019), 505–530. Google Scholar | DOI

[27] Dai, X. and Martone, G., ‘Correlation of the renormalized Hilbert length for convex projective surfaces’, Ergodic Theory Dynam. Systems 43 (2023), 2938–2973. Google Scholar

[28] Dal’Bo, F., ‘Remarques sur le spectre des longueurs d’une surface et comptages’, Bol. Soc. Brasil. Mat. (N.S.) 30(2) (1999), 199–221. Google Scholar | DOI

[29] Dembo, A. and Zeitouni, O., Large Deviations Techniques and Applications . Applications of Mathematics (New York), vol. 38, 2nd ed. (Springer-Verlag, New York, 1998).10.1007/978-1-4612-5320-4 Google Scholar | DOI

[30] Druţu, C. and Sapir, M., ‘Tree-graded spaces and asymptotic cones of groups’. With an appendix by D. V. Osin and M. Sapir. Topology 44 (2005), 959–1058. Google Scholar

[31] Erlandsson, V., ‘A remark on the word length in surface groups’. Trans. Amer. Math. Soc. 327 (2019), 441–455.10.1090/tran/7561 Google Scholar | DOI

[32] Epstein, D. B. A., Cannon, J. W., Holt, D., Levy, S., Paterson, M. and Thurston, W., Word Processing in Groups (Jones and Bartlett Publishers, Boston, MA, 1992). Google Scholar | DOI

[33] Farb, B. and Margalit, D., A Primer on Mapping Class Groups . Princeton Mathematical Series, vol. 49 (Princeton University Press, 2012). Google Scholar

[34] Fioravanti, E., ‘On automorphisms and splittings of special groups’, Compos. Math. 159(2) (2023), 232–305.10.1112/S0010437X22007850 Google Scholar | DOI

[35] Fioravanti, E., Levcovitz, I. and Sageev, M., ‘Coarse cubical rigidity’, J. Topol. 17(3) (2024), Paper No. e12353, 50.10.1112/topo.12353 Google Scholar | DOI

[36] Gekhtman, I., Taylor, S. J. and Tiozzo, G., ‘Central limit theorems for counting measures in coarse negative curvature’, Compos. Math. 158(10) (2022), 1980–2013. Google Scholar | DOI

[37] Gekhtman, I., Taylor, S. J. and Tiozzo, G., ‘Counting loxodromics for hyperbolic actions’, J. Topol. 11(2) (2018), 379–419.10.1112/topo.12053 Google Scholar | DOI

[38] Gekhtman, I., Taylor, S. J. and Tiozzo, G., ‘Counting problems in graph products and relatively hyperbolic groups’, Israel J. Math. 237 (2020), 311–371.10.1007/s11856-020-2008-x Google Scholar | DOI

[39] Genevois, A., ‘Hyperbolicities in CAT(0) cube complexes’, Enseign. Math. 65(1/2) (2019), 33–100.10.4171/lem/65-1/2-2 Google Scholar | DOI

[40] Godelle, E. and Paris, L., ‘K(π,1) and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups’, Math. Z. 272 (2012), 1339–1364.10.1007/s00209-012-0989-9 Google Scholar | DOI

[41] Groves, D. and Manning, J. F., ‘Specializing cubulated relatively hyperbolic groups’, J. Topol. 15 (2022), 398–442. Google Scholar | DOI

[42] Guivarc’H, Y. and Hardy, J., ‘Théorèmes limites pour une classe de chaines de Markov et applications aux difféomorphismes d’Anosov’, Ann. Inst. Henri Poincaré Probab. Stat. 24(1) (1988), 73–98. Google Scholar

[43] Hagen, M. F. and Przytycki, P., ‘Cocompactly cubulated graph manifolds’, Israel J. Math. 207(1) (2015), 377–394. Google Scholar | DOI

[44] Hagen, M. F. and Wise, D. T., ‘Cubulating hyperbolic free-by-cyclic groups: the general case’, Geom. Funct. Anal. 25 (2015), 134–179. Google Scholar

[45] Hagen, M. F. and Wise, D. T., ‘Cubulating hyperbolic free-by-cyclic groups: the irreducible case’, Duke Math. J. 165(9) (2016), 1753–1813.10.1215/00127094-3450752 Google Scholar | DOI

[46] Haglund, F., ‘Isometries of CAT(0) cube complexes are semi-simple’, Ann. Math. Québec (2021). https://doi.org/10.1007/s40316-021-00186-2. Google Scholar

[47] Haglund, F. and Wise, D. T., ‘Special cube complexes’, Geom. Funct. Anal. 17(5) (2008), 1551–1620.10.1007/s00039-007-0629-4 Google Scholar | DOI

[48] He, Y. M., Lee, H. and Park, I., Pressure metrics in geometry and dynamics. , 2024. Google Scholar | arXiv

[49] Hermiller, S. and Meier, J., ‘Algorithms and geometry for graph products of groups’, J. Algebra 171(1) (1995), 230–257.10.1006/jabr.1995.1010 Google Scholar | DOI

[50] Hruska, G. C., ‘Relative hyperbolicity and relative quasiconvexity for countable groups’, Algebr. Geom. Topol. 10(3) (2010), 1807–1856.10.2140/agt.2010.10.1807 Google Scholar | DOI

[51] Hurwitz, A., ‘Über die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche’, Math. Ann. 39(2) (1891), 279–284.10.1007/BF01206656 Google Scholar | DOI

[52] Kahn, J. and Markovic, V., ‘Immersing almost geodesic surfaces in a closed hyperbolic three manifold’, Ann. Math. (2) 175(3) (2012), 1127–1190. Google Scholar | DOI

[53] Kao, L.-Y., ‘Entropy rigidity, pressure metric, and immersed surfaces in hyperbolic 3-Manifolds’. In Thermodynamic Formalism: CIRM Jean-Morlet Chair, Fall 2019 (Springer, International Publishing, 2021), pp. 351–393. Google Scholar | DOI

[54] Kao, L.-Y., ‘Manhattan curves for hyperbolic surfaces with cusps’, Ergodic Theory Dynam. Systems 40(7) (2020), 1843–1874.10.1017/etds.2018.124 Google Scholar | DOI

[55] Kao, L.-Y., ‘Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces’, Israel J. Math. 240(2) (2020), 567–602.10.1007/s11856-020-2073-1 Google Scholar | DOI

[56] Kifer, Y., ‘Large deviations, averaging and periodic orbits of dynamical systems’, Comm. Math. Phys. 162 (1994), 33–46.10.1007/BF02105185 Google Scholar | DOI

[57] Lauer, J. and Wise, D. T., ‘Cubulating one-relator groups with torsion’, Math. Proc. Cambridge Philos. Soc. 155(3) (2013), 411–429.10.1017/S0305004113000285 Google Scholar | DOI

[58] Li, J. and Wise, D. T., ‘No growth-gaps for special cube complexes’, Groups Geom. Dyn. 14(1) (2020), 117–135. Google Scholar | DOI

[59] Marcus, B. and Tuncel, S., ‘Entropy at a weight-per-symbol and embeddings of Markov chains’, Invent. Math. 102 (1990), 235–266. Google Scholar | DOI

[60] Martin, A. and Steenbock, M., ‘A combination theorem for cubulation in small cancellation theory over free products’, Ann. Inst. Fourier (Grenoble) 67 (2017), 1613–1670.10.5802/aif.3118 Google Scholar | DOI

[61] Mcmullen, C., ‘Thermodynamics, dimension and the Weil–Petersson metric’, Invent. Math. 173 (2008), 365–425.10.1007/s00222-008-0121-2 Google Scholar | DOI

[62] Niblo, G. A. and Reeves, L. D., ‘Coxeter groups act on CAT(0) cube complexes’, J. Group Theory 6(3) (2003), 399–413.10.1515/jgth.2003.028 Google Scholar | DOI

[63] Odrzygóźdź, T., ‘Cubulating random groups in the square model’, Israel J. Math. 227(2) (2018), 623–661.10.1007/s11856-018-1734-9 Google Scholar | DOI

[64] Ollivier, Y. and Wise, D. T., ‘Cubulating random groups at density less than 1/6’, Trans. Amer. Math. Soc. 363(9) (2011), 4701–4733.10.1090/S0002-9947-2011-05197-4 Google Scholar | DOI

[65] Parry, W., ‘Intrinsic Markov chains, Trans. Amer. Math. Soc. 112 (1964), 55–66.10.1090/S0002-9947-1964-0161372-1 Google Scholar | DOI

[66] Parry, W. and Pollicott, M., ‘Zeta functions and the periodic orbit structure of hyperbolic dynamics’, Astérisque 187–188 (1990), 1–268. Google Scholar

[67] Pollicott, M. and Sharp, R., ‘A Weil–Petersson metric on spaces of metric graphs’, Geom. Dedicata 172 (2014), 229–244. Google Scholar | DOI

[68] Pollicott, M. and Sharp, R., ‘Large deviations, fluctuations and shrinking intervals’, Comm. Math. Phys. 290 (2009), 321–334.10.1007/s00220-008-0725-9 Google Scholar | DOI

[69] Przytycki, P. and Wise, D. T., ‘Graph manifolds with boundary are virtually special’, J. Topol. 7(2) (2014), 419–435.10.1112/jtopol/jtt009 Google Scholar | DOI

[70] Przytycki, P. and Wise, D. T., ‘Mixed 3-manifolds are virtually special’, J. Amer. Math. Soc. 31(2) (2018), 319–347. Google Scholar

[71] Reyes, E., ‘On cubulated relatively hyperbolic groups’, Geom. Topol. 27 (2023), 575–640.10.2140/gt.2023.27.575 Google Scholar | DOI

[72] Roller, M. A., Poc sets, median algebras and group actions. An extended study of Dunwoody’s construction and Sageev’s theorem. ; University of Southampton preprint, 1998. Google Scholar | arXiv

[73] Rousseau-Egele, J., ‘Un théoreme de la limite locale pour une classe de transformations dilatantes et monotones par morceaux’, Ann. Probab. 11 (1983), 772–788.10.1214/aop/1176993522 Google Scholar | DOI

[74] Sageev, M., ‘CAT(0) cube complexes and groups’. In Geometric Group Theory, IAS/Park City Math. Ser., vol. 21 (American Mathematical Society, Providence, RI, 2014), pp. 7–54. Google Scholar | DOI

[75] Sageev, M., ‘Ends of group pairs and non-positively curved cube complexes’, Proc. Lond. Math. Soc. 71(3) (1995), 585–617.10.1112/plms/s3-71.3.585 Google Scholar | DOI

[76] Sageev, M. and Wise, D. T., ‘Cores for quasiconvex actions’, Proc. Amer. Math. Soc. 143 (2015), 2731–2741.10.1090/S0002-9939-2015-12297-6 Google Scholar | DOI

[77] Sale, A. and Susse, T., ‘Outer automorphism groups of right-angled Coxeter groups are either large or virtually abelian’, Trans. Amer. Math. Soc. 372(11) (2019), 7785–7803.10.1090/tran/7897 Google Scholar | DOI

[78] Schwartz, R. and Sharp, R., ‘The correlation of length spectra of two hyperbolic surfaces’, Comm. Math. Phys. 154 (1993), 423–430. Google Scholar | DOI

[79] Sharp, R., ‘Comparing length functions on free groups’. In Spectrum and Dynamics, CRM Conf. Proc. Lecture Notes, vol. 52 (Amer. Math. Soc., Providence, RI, 2010), pp. 185–207.10.1090/crmp/052/10 Google Scholar | DOI

[80] Sharp, R., ‘Distortion and entropy for automorphisms of free groups’, Discrete Contin. Dyn. Syst. 26 (2010), 347–363.10.3934/dcds.2010.26.347 Google Scholar | DOI

[81] Sharp, R., ‘The Manhattan curve and the correlation of length spectra on hyperbolic surfaces’, Math. Z. 228 (1998), 745–750.10.1007/PL00004643 Google Scholar | DOI

[82] Sigmund, K., ‘Generic properties of invariant measures for Axiom A diffeomorphisms’, Invent. Math. 11 (1970), 99–109.10.1007/BF01404606 Google Scholar | DOI

[83] Stucky, B., ‘Cubulating one-relator products with torsion’, Groups Geom. Dyn. 15 (2021), 691–754.10.4171/ggd/619 Google Scholar | DOI

[84] Tidmore, J., ‘Cocompact cubulations of mixed 3-manifolds’, Groups Geom. Dyn. 12(4) (2018), 1429–1460.10.4171/ggd/474 Google Scholar | DOI

[85] Wise, D. T., ‘Cubulating small cancellation groups’, Geom. Funct. Anal. 14(1) (2004), 150–214.10.1007/s00039-004-0454-y Google Scholar | DOI

[86] Wise, D. T., ‘The structure of groups with a quasiconvex hierarchy’, Ann. of Math. Stud. 209 (2021), 1–376. Google Scholar

[87] Wise, D. T. and Woodhouse, D. J., ‘A cubical flat torus theorem and the bounded packing property’, Israel J. Math. 217 (2017), 263–281.10.1007/s11856-017-1445-7 Google Scholar | DOI

[88] Yang, W., ‘Statistically convex-cocompact actions of groups with contracting elements’, Int. Math. Res. Not. IMRN 2019(23), 7259–7323. Google Scholar

Cité par Sources :