The homology of the groupoid of the self-similar infinite dihedral group


  • Eduard Ortega
  • Alvaro Sanchez



We compute the $K$-theory of the $C^*$-algebra associated to the self-similar infinite dihedral group, and the homology of its associated étale groupoid. We see that the rational homology differs from the $K$-theory, strongly contradicting a conjecture posted by Matui. Moreover, we compute the abelianization of the topological full group of the groupoid associated to the self-similar infinite dihedral group.


Exel, R., and Pardo, E., Self-similar graphs, a unified treatment of Katsura and Nekrashevych $C^*$-algebras, Adv. Math. 306 (2017), 1046–1129.

Farsi, C., Kumjian, A., Pask, D., and Sims, A., Ample groupoids: equivalence, homology, and Matui's HK conjecture, Münster J. Math. 12 (2019), no. 2, 411–451.

Giordano, T., Putnam, I. F., and Skau, C. F., Full groups of Cantor minimal systems, Israel J. Math 111 (1999), 285–320.

Grigorchuck, R., Nekrashevych, V., and Sunic, Z., From self-similar groups to self-similar spectra, Fractal geometry and stochastics V, 175–207, Progr. Probab., 70, Birkhäuser/Springer, Cham, 2015.

Juschenko K., and Monod, N., Cantor systems, piecewise translations and simple amenable groups, Ann. of Math. (2), 178 (2013), no. 2, 775–787.

Matui, H., Homology and topological full groups of étale groupoids on totally disconnected spaces, Proc. London Math. Soc. (3) 104 (2012), no. 1, 27–56.

Matui, H., Topological full groups of one-sided shifts of finite type, J. Reine Angew. Math. 705 (2015), 35–84.

Matui, H., Étale groupoids arising from products of shifts of finite type, Adv. Math. 303 (2016), 502–548.

Nekrashevych, V., $C^*$-algebras and self-similar groups. J. Reine Angew. Math. 630 (2009), 59–123.

Nyland, P., and Ortega, E., Katsura-Exel-Pardo groupoids and the AH conjecture, arXiv:2007.06638.

Ortega, E., The homology of the Katsura-Exel-Pardo groupoids., J. Noncommut. Geom. 14 (2020), no. 3, 1–23.

Proietti, V., and Yamashita, M., Homology and K-theory of torsion-free ample groupoids and Smale spaces, arXiv:2006.08028.

Scarparo, E., Homology of odometers, Ergodic Theory Dynam. Systems 40 (2020), no. 9, 2541–2551.

Weibel, C. A., An introduction to homological algebra, Cambridge University Press, Cambridge, 1994.



How to Cite

Ortega, E., & Sanchez, A. (2022). The homology of the groupoid of the self-similar infinite dihedral group: Array. MATHEMATICA SCANDINAVICA, 128(2).