# American Institute of Mathematical Sciences

August  2013, 33(8): 3753-3765. doi: 10.3934/dcds.2013.33.3753

## K-groups of the full group actions on one-sided topological Markov shifts

 1 Department of Mathematics, Joetsu University of Education, Joetsu 943-8512, Japan

Received  May 2012 Revised  July 2012 Published  January 2013

Let $(X_A,\sigma_A)$ be the one-sided topological Markov shift for an $N\times N$ irreducible matrix $A$ with entries in $\{ 0,1 \}$. We will first show that the continuous full group $\Gamma_A$ of $(X_A,\sigma_A)$ is non amenable as a discrete group and contains all finite groups and free groups. We will second introduce the K-group $K^0(X_A,\Gamma_A)$ for the action of $\Gamma_A$ on $X_A$, and show that the group $K^0(X_A,\Gamma_A)$ with the position of the constant $1$ function is a complete invariant for the topological conjugacy class of the action of $\Gamma_A$ on $X_A$ under some conditions.
Citation: Kengo Matsumoto. K-groups of the full group actions on one-sided topological Markov shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3753-3765. doi: 10.3934/dcds.2013.33.3753
##### References:
 [1] M. Boyle, "Topological Orbit Equivalence and Factor Maps in Symbolic Dynamics," Ph.D. thesis, University of Washington, 1983. [2] M. Boyle and D. Handelman, Orbit equivalence, flow equivalence and ordered cohomology, Israel J. Math., 95 (1996), 169-210. doi: 10.1007/BF02761039. [3] M. D. Choi, A simple $C^*$-algebras generated by two finite order unitaries, Canad. J. Math., 31 (1979), 867-880. doi: 10.4153/CJM-1979-082-4. [4] J. Cuntz, Simple $C^*$-algebras generated by isometries, Comm. Math. Phys., 57 (1977), 173-185. [5] J. Cuntz, A class of $C^*$-algebras and topological Markov chains. II. Reducible chains and the Ext-functor for $C^*$-algebras, Invent. Math., 63 (1981), 25-40. doi: 10.1007/BF01389192. [6] J. Cuntz, K-theory for certain $C^*$-algebras, Ann. Math. (2), 113 (1981), 181-197. doi: 10.2307/1971137. [7] J. Cuntz and W. Krieger, A class of $C^*$-algebras and topological Markov chains, Invent. Math., 56 (1980), 251-268. doi: 10.1007/BF01390048. [8] T. Giordano, I. F. Putnam and C. F. Skau, Topological orbit equivalence and $C^*$-crossed products, J. Reine Angew. Math., 469 (1995), 51-111. [9] T. Giordano, I. F. Putnam and C. F. Skau, Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320. doi: 10.1007/BF02810689. [10] T. Giordano, H. Matui, I. F. Putnam and C. F. Skau, Orbit equivalence for Cantor minimal $\mathbb Z^d$-systems, Invent. Math., 179 (2010), 119-158. doi: 10.1007/s00222-009-0213-7. [11] R. H. Herman, I. F. Putnam and C. F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math., 3 (1992), 827-864. doi: 10.1142/S0129167X92000382. [12] W. Magnus, A. Karrass and D. Solitar, "Combinatorial Group Theory. Presentations of groups in terms of generators and relations," Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966. [13] K. Matsumoto, On automorphisms of $C^*$-algebras associated with subshifts, J. Operator Theory, 44 (2000), 91-112. [14] K. Matsumoto, Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Pacific J. Math., 246 (2010), 199-225. doi: 10.2140/pjm.2010.246.199. [15] K. Matsumoto, Classification of Cuntz-Krieger algebras by orbit equivalence of topological Markov shifts, to appear in Proc. Amer. Math. Soc. [16] K. Matsumoto, Full groups of one-sided topological Markov shifts, preprint, arXiv:1205.1320. [17] H. Matui, Homology and topological full groups of étale groupoids on totally disconnected spaces, Proc. London Math. Soc. (3), 104 (2012), 27-56. doi: 10.1112/plms/pdr029. [18] I. F. Putnam, The $C^*$-algebras associated with minimal homeomorphisms of the Cantor set, Pacific J. Math., 136 (1989), 329-353. [19] M. Rφrdam, Classification of Cuntz-Krieger algebras, K-theory, 9 (1995), 31-58. doi: 10.1007/BF00965458. [20] M. Rφrdam, F. Larsen and N. J. Laustsen, "An Introduction to K-Theory for $C^*$-Algebras," London Mathematical Society Student Texts, 49, Cambridge University Press, Cambridge, 2000. [21] J. Spielberg, Free-product groups, Cuntz-Krieger algebras, and covariant maps, Internat. J. Math., 2 (1991), 457-476. doi: 10.1142/S0129167X91000260.

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##### References:
 [1] M. Boyle, "Topological Orbit Equivalence and Factor Maps in Symbolic Dynamics," Ph.D. thesis, University of Washington, 1983. [2] M. Boyle and D. Handelman, Orbit equivalence, flow equivalence and ordered cohomology, Israel J. Math., 95 (1996), 169-210. doi: 10.1007/BF02761039. [3] M. D. Choi, A simple $C^*$-algebras generated by two finite order unitaries, Canad. J. Math., 31 (1979), 867-880. doi: 10.4153/CJM-1979-082-4. [4] J. Cuntz, Simple $C^*$-algebras generated by isometries, Comm. Math. Phys., 57 (1977), 173-185. [5] J. Cuntz, A class of $C^*$-algebras and topological Markov chains. II. Reducible chains and the Ext-functor for $C^*$-algebras, Invent. Math., 63 (1981), 25-40. doi: 10.1007/BF01389192. [6] J. Cuntz, K-theory for certain $C^*$-algebras, Ann. Math. (2), 113 (1981), 181-197. doi: 10.2307/1971137. [7] J. Cuntz and W. Krieger, A class of $C^*$-algebras and topological Markov chains, Invent. Math., 56 (1980), 251-268. doi: 10.1007/BF01390048. [8] T. Giordano, I. F. Putnam and C. F. Skau, Topological orbit equivalence and $C^*$-crossed products, J. Reine Angew. Math., 469 (1995), 51-111. [9] T. Giordano, I. F. Putnam and C. F. Skau, Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320. doi: 10.1007/BF02810689. [10] T. Giordano, H. Matui, I. F. Putnam and C. F. Skau, Orbit equivalence for Cantor minimal $\mathbb Z^d$-systems, Invent. Math., 179 (2010), 119-158. doi: 10.1007/s00222-009-0213-7. [11] R. H. Herman, I. F. Putnam and C. F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math., 3 (1992), 827-864. doi: 10.1142/S0129167X92000382. [12] W. Magnus, A. Karrass and D. Solitar, "Combinatorial Group Theory. Presentations of groups in terms of generators and relations," Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966. [13] K. Matsumoto, On automorphisms of $C^*$-algebras associated with subshifts, J. Operator Theory, 44 (2000), 91-112. [14] K. Matsumoto, Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Pacific J. Math., 246 (2010), 199-225. doi: 10.2140/pjm.2010.246.199. [15] K. Matsumoto, Classification of Cuntz-Krieger algebras by orbit equivalence of topological Markov shifts, to appear in Proc. Amer. Math. Soc. [16] K. Matsumoto, Full groups of one-sided topological Markov shifts, preprint, arXiv:1205.1320. [17] H. Matui, Homology and topological full groups of étale groupoids on totally disconnected spaces, Proc. London Math. Soc. (3), 104 (2012), 27-56. doi: 10.1112/plms/pdr029. [18] I. F. Putnam, The $C^*$-algebras associated with minimal homeomorphisms of the Cantor set, Pacific J. Math., 136 (1989), 329-353. [19] M. Rφrdam, Classification of Cuntz-Krieger algebras, K-theory, 9 (1995), 31-58. doi: 10.1007/BF00965458. [20] M. Rφrdam, F. Larsen and N. J. Laustsen, "An Introduction to K-Theory for $C^*$-Algebras," London Mathematical Society Student Texts, 49, Cambridge University Press, Cambridge, 2000. [21] J. Spielberg, Free-product groups, Cuntz-Krieger algebras, and covariant maps, Internat. J. Math., 2 (1991), 457-476. doi: 10.1142/S0129167X91000260.
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