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The large scale geometry of strongly aperiodic subshifts of finite type. (English) Zbl 1400.20034
Summary: A subshift on a group $$G$$ is a closed, $$G$$-invariant subset of $$A^G$$, for some finite set $$A$$. It is said to be a subshift of finite type (SFT) if it is defined by a finite collection of “forbidden patterns”, to be strongly aperiodic if all point stabilizers are trivial, and weakly aperiodic if all point stabilizers are infinite index in $$G$$. We show that groups with at least 2 ends have a strongly aperiodic SFT, and that having such an SFT is a QI invariant for finitely presented groups. We show that a finitely presented torsion free group with no weakly aperiodic SFT must be QI-rigid. The domino problem on $$G$$ asks whether the SFT specified by a given set of forbidden patterns is empty. We show that decidability of the domino problem is a QI invariant.

##### MSC:
 20F65 Geometric group theory 37B10 Symbolic dynamics 37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) 20F05 Generators, relations, and presentations of groups
##### Keywords:
geometric group theory; symbolic dynamics
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##### References:
 [1] Aubrun, N.; Kari, J., Tiling problems on Baumslag-Solitar groups, (MCU, (2013)), 35-46 [2] Ballier, A.; Stein, M., The domino problem on groups of polynomial growth, (2013), arXiv preprint [3] Berger, R., The undecidability of the domino problem, Mem. Amer. Math. Soc., vol. 66, (1966) · Zbl 0199.30802 [4] Bestvina, M.; Brady, N., Morse theory and finiteness properties of groups, Invent. Math., 129, 3, 445-470, (1997) · Zbl 0888.20021 [5] Block, J.; Weinberger, S., Aperiodic tilings, positive scalar curvature, and amenability of spaces, J. Amer. Math. Soc., 5, 4, 907-918, (1992) · Zbl 0780.53031 [6] Carroll, D.; Penland, A., Periodic points on shifts of finite type and commensurability invariants of groups, (2015), arXiv preprint · Zbl 1360.37037 [7] Ceccherini-Silberstein, T.; Coornaert, M., Cellular automata and groups, (2009), Springer · Zbl 1160.37317 [8] Coornaert, M.; Papadopoulos, A., Symbolic dynamics and hyperbolic groups, (1993), Springer · Zbl 0783.58017 [9] Y. Cornulier, personal communication. [10] Culik, K. C.; Kari, J., An aperiodic set of Wang cubes, J.UCS, 675-686, (1996) · Zbl 0942.68595 [11] Epstein, D.; Paterson, M. S.; Cannon, J. W.; Holt, D. F.; Levy, S. V.; Thurston, W. P., Word processing in groups, (1992), AK Peters, Ltd. · Zbl 0764.20017 [12] Hopf, H., Enden offener räume und unendliche diskontinuierliche gruppen, Comment. Math. Helv., 16, 1, 81-100, (1943) · Zbl 0060.40008 [13] Jeandel, E., Some notes about subshifts on groups, (2015), arXiv preprint [14] Jeandel, E., Translation-like actions and aperiodic subshifts on groups, (2015), arXiv preprint [15] Jeandel, E.; Theyssier, G., Subshifts, languages and logic, (Developments in Language Theory, (2009), Springer), 288-299 · Zbl 1247.03068 [16] Kuske, D.; Lohrey, M., Logical aspects of Cayley-graphs: the group case, Ann. Pure Appl. Logic, 131, 1, 263-286, (2005) · Zbl 1063.03005 [17] Marcinkowski, M.; Nowak, P. W., Aperiodic tilings of manifolds of intermediate growth, (2012), arXiv preprint · Zbl 1339.53033 [18] Mozes, S., Aperiodic tilings, Invent. Math., 128, 3, 603-611, (1997) · Zbl 0879.52011 [19] Muller, D. E.; Schupp, P. E., Groups, the theory of ends, and context-free languages, J. Comput. System Sci., 26, 3, 295-310, (1983) · Zbl 0537.20011 [20] Muller, D. E.; Schupp, P. E., The theory of ends, pushdown automata, and second-order logic, Theoret. Comput. Sci., 37, 51-75, (1985) · Zbl 0605.03005 [21] Piantadosi, S. T., Symbolic dynamics on free groups, Discrete Contin. Dyn. Syst., 20, 3, 725, (2008) · Zbl 1140.37006 [22] Robinson, R. M., Undecidability and nonperiodicity for tilings of the plane, Invent. Math., 12, 3, 177-209, (1971) · Zbl 0197.46801 [23] A. Şahin, M. Schraudner, I. Ugarcovici, Strongly aperiodic shifts of finite type for the heisenberg group (preliminary title), 2014. [24] Stallings, J. R., On torsion-free groups with infinitely many ends, Ann. of Math., 312-334, (1968) · Zbl 0238.20036
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