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Computing the fundamental group of a higher-rank graph

Part of: Categories with structure (Co)homology theory

Published online by Cambridge University Press:  26 August 2021

Sooran Kang ,
David Pask  and
Samuel B.G. Webster
Sooran Kang
Affiliation:
College of General Education, Chung-Ang University, Seoul 06974, Republic of Korea (sooran09@cau.ac.kr)
David Pask
Affiliation:
School of Mathematics and Applied Statistics, The University of Wollongong, Wollongong, NSW 2522, Australia (dpask@uow.edu.au)
Samuel B.G. Webster
Affiliation:
Independent Hospital Pricing Authority, Level 6, 1 Oxford Street, Sydney, NSW 2000, Australia (sbgwebster@gmail.com)
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Abstract

We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental groups of several examples from the literature. Our results fit naturally into the suite of known geometrical results about higher-rank graphs when we show that the abelianization of the fundamental group is the homology group. We end with a calculation which gives a non-standard presentation of the fundamental group of the Klein bottle to the one normally found in the literature.

Keywords

Higher-rank graphfundamental group

MSC classification

Primary: 14F35: Homotopy theory; fundamental groups
Secondary: 18D99: None of the above, but in this section

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 3 , August 2021 , pp. 650 - 661
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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References

Aranda-Pino, G., Clark, J., an Huef, A. and Raeburn, I., Kumjian-Pask algebras of higher-rank graphs, Trans. Am. Math. Soc., 365 (2013), 3613–3641.CrossRefGoogle Scholar
Carlsen, T., Kang, S., Shotwell, J. and Sims, A., The primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources, J. Funct. Anal. 266 (2014), 2570–2589.CrossRefGoogle Scholar
Clark, L. O., Flynn, C. and an Huef, A., Kumjian-Pask algebras of locally convex higher-rank graphs, J. Algebra 399 (2014), 445–474.CrossRefGoogle Scholar
Evans, D. G., On the K-theory of higher-rank graph C*-algebras, N. Y. J. Math. 14 (2008), 1–31.Google Scholar
Farsi, C., Gillaspy, E., Jorgensen, P. E. T., Kang, S. and Packer, J. A., Purely atomic representations of higher-rank graph $C^{*}$-algebras, Int. Eq. Oper. Theory 90 (2018), paper 67, 26 pages.Google Scholar
Farthing, C., Pask, D. and Sims, A., Crossed products of $k$-graph $C^{\ast }$-algebras by $\mathbb {Z}^{l}$, Houston J. Math. 35 (2009), 903–933.Google Scholar
an Huef, A., Laca, M., Raeburn, I. and Sims, A., KMS states on $C^{*}$-algebras associated to higher-rank graphs, J. Math. Anal. Appl. 405 (2013), 388–399.CrossRefGoogle Scholar
Hazlewood, R., Raeburn, I., Sims, A. and Webster, S. B. G., Remarks on some fundamental results about higher-rank graphs and their $C^{*}$-algebras, Proc. Edinburgh Math. Soc. 56 (2013), 575–597.CrossRefGoogle Scholar
Kaliszewski, S., Kumjian, A., Quigg, J. and Sims, A., Topological realizations and fundamental groups of higher-rank graphs, Proc. Edinb. Math. Soc. 59 (2016), 143–168.CrossRefGoogle Scholar
Kimberley, J. and Robertson, G., Groups acting on trees, tiling systems and analytic K-Theory, New York J. Math 8 (2002), 111–131.Google Scholar
Kumjian, A. and Pask, D., Higher-rank graph $C^{*}$-algebras, New York J. Math 6 (2000), 1–20.Google Scholar
Kumjian, A., Pask, D. and Sims, A., Homology for higher-rank graphs and twisted $C^{*}$-algebras, J. Funct. Anal. 263 (2012), 1539–1574.CrossRefGoogle Scholar
Kumjian, A., Pask, D. and Sims, A., On the $K$-theory of twisted higher-rank graph $C^{*}$-algebras, J. Math. Anal. Appl. 401 (2013), 104–113.CrossRefGoogle Scholar
Kumjian, A., Pask, D. and Sims, A., On twisted higher-rank graph $C^{*}$-algebras, Trans. Am. Math. Soc. 367 (2015), 5177–5216.CrossRefGoogle Scholar
Massey, W., A basic course in algebraic topology, Graduate Texts in Mathematics, Volume 127 (Springer-Verlag, Berlin/New York, 1991).CrossRefGoogle Scholar
Mutter, S., The K-Theory of 2-rank graphs associated to complete bipartite graphs. arXiv: 2004.11602v3Google Scholar
Mutter, S., Radu, A-C. and Vdovina, A., $C^{*}$-algebras of higher rank graphs from groups acting on buildings, and explicit computation of their K-theory. arXiv: 2012.05561v1Google Scholar
Pask, D., Quigg, J. and Raeburn, I., Fundamental groupoids of $k$-graphs, N. Y. J. Math. 10 (2004), 195–207.Google Scholar
Raeburn, I., Graph algebras, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2005, vi+113.CrossRefGoogle Scholar
Raeburn, I., Sims, A. and Yeend, T., Higher-rank graphs and their $C^{*}$-algebras, Proc. Edinb. Math. Soc. 46 (2003), 99–115.CrossRefGoogle Scholar
Robertson, G. and Steger, T., Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras, J. Reine Angew. Math. 513 (1999), 115–144.CrossRefGoogle Scholar
Stillwell, J. C., Classical topology and combinatorial group theory. Graduate Texts in Mathematics, Volume 72 (Springer-Verlag, New York-Berlin, 1980)CrossRefGoogle Scholar
Yang, D., Endomorphisms and modular theory of $2$-graph $C^{*}$-algebras, Indiana Univ. Math. J. 59 (2010), 495–520.CrossRefGoogle Scholar