Revista de la
Unión Matemática Argentina
Strongly aperiodic SFTs on hyperbolic groups: where to find them and why we love them
Yo'av Rieck
Volume 64, no. 2 (2023), pp. 355–373    

https://doi.org/10.33044/revuma.3155

Download PDF

Abstract

D. B. Cohen, C. Goodman-Strauss, and the author [Ergodic Theory Dynam. Systems 42 (2022), no. 9, 2740–2783] proved that a hyperbolic group admits an SA SFT if and only if it has at most one end. This paper has two distinct parts: the first is a conversation explaining what an SA SFT is and how it may be of use, while in the second part I attempt to explain both old and new ideas that go into that proof. References to specific claims in the work cited above are given, with the hope that any interested reader may be able to find the details there more accesible after reading this exposition.

References

  1. N. Aubrun, S. Barbieri, and E. Jeandel, About the domino problem for subshifts on groups, in Sequences, Groups, and Number Theory, 331–389, Trends Math, Birkhäuser/Springer, Cham, 2018. MR 3799931.
  2. N. Aubrun, S. Barbieri, and E. Moutot, The domino problem is undecidable on surface groups, in 44th International Symposium on Mathematical Foundations of Computer Science, Art. 46, 14 pp., LIPIcs. Leibniz Int. Proc. Inform., 138, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2019. MR 4008435.
  3. N. Aubrun and J. Kari, Tiling problems on Baumslag-Solitar groups, in Proceedings: Machines, Computations and Universality 2013, 35–46, Electron. Proc. Theor. Comput. Sci. (EPTCS), 128, EPTCS, 2013. MR 3594268.
  4. N. Aubrun and J. Kari, On the domino problem of the Baumslag-Solitar groups, Theoret. Comput. Sci. 894 (2021), 12–22. MR 4334698.
  5. S. Barbieri, A geometric simulation theorem on direct products of finitely generated groups, Discrete Anal. 2019, Paper No. 9, 25 pp. MR 3964145.
  6. S. Barbieri, M. Sablik, and V. Salo, Groups with Self-Simulable Zero-Dimensional Dynamics, arXiv:2104.05141 [math.GR], 2021.
  7. M. R. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Grundlehren der mathematischen Wissenschaften, 319, Springer-Verlag, Berlin, 1999. MR 1744486.
  8. D. B. Cohen, The large scale geometry of strongly aperiodic subshifts of finite type, Adv. Math. 308 (2017), 599–626. MR 3600067.
  9. D. B. Cohen and C. Goodman-Strauss, Strongly aperiodic subshifts on surface groups, Groups Geom. Dyn. 11 (2017), no. 3, 1041–1059. MR 3692905.
  10. D. B. Cohen, C. Goodman-Strauss, and Y. Rieck, Strongly aperiodic subshifts of finite type on hyperbolic groups, Ergodic Theory Dynam. Systems 42 (2022), no. 9, 2740–2783. MR 4461690.
  11. M. Coornaert and A. Papadopoulos, Symbolic Dynamics and Hyperbolic Groups, Lecture Notes in Mathematics, 1539, Springer-Verlag, Berlin, 1993. MR 1222644.
  12. M. Coornaert and A. Papadopoulos, Horofunctions and symbolic dynamics on Gromov hyperbolic groups, Glasg. Math. J. 43 (2001), no. 3, 425–456. MR 1878587.
  13. F. Dahmani, D. Futer, and D. T. Wise, Growth of quasiconvex subgroups, Math. Proc. Cambridge Philos. Soc. 167 (2019), no. 3, 505–530. MR 4015648.
  14. F. Dahmani and V. Guirardel, The isomorphism problem for all hyperbolic groups, Geom. Funct. Anal. 21 (2011), no. 2, 223–300. MR 2795509.
  15. D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, and W. P. Thurston, Word Processing in Groups, Jones and Bartlett, Boston, MA, 1992. MR 1161694.
  16. J. Esnay and E. Moutot, Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups arXiv:2004.02534 [math.DS], 2021.
  17. C. Goodman-Strauss, A strongly aperiodic set of tiles in the hyperbolic plane, Invent. Math. 159 (2005), no. 1, 119–132. MR 2142334.
  18. M. Gromov, Hyperbolic groups, in Essays in Group Theory, 75–263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987. MR 0919829.
  19. E. Jeandel, Aperiodic Subshifts of Finite Type on Groups, arXiv:1501.06831v2 [math.GR], 2015.
  20. J. Kari, A small aperiodic set of Wang tiles, Discrete Math. 160 (1996), no. 1-3, 259–264. MR 1417578.
  21. S. Mozes, Aperiodic tilings, Invent. Math. 128 (1997), no. 3, 603–611. MR 1452434.
  22. C. Radin, The pinwheel tilings of the plane, Ann. of Math. (2) 139 (1994), no. 3, 661–702. MR 1283873.
  23. Z. Sela, The isomorphism problem for hyperbolic groups. I, Ann. of Math. (2) 141 (1995), no. 2, 217–283. MR 1324134.
  24. B. Seward, Burnside's Problem, spanning trees and tilings, Geom. Topol. 18 (2014), no. 1, 179–210. MR 3158775.
  25. G. A. Swarup, On the cut point conjecture, Electron. Res. Announc. Amer. Math. Soc. 2 (1996), no. 2, 98–100. MR 1412948.
  26. H. Wang, Computation, Logic, Philosophy, Mathematics and its Applications (Chinese Series), 2, Science Press Beijing, Beijing, 1990. MR 1112395.