K-groups of the full group actionson  one-sided topological Markov shifts

Kengo Matsumoto

doi:10.3934/dcds.2013.33.3753

Article Contents

2013, Volume 33, Issue 8: 3753-3765. Doi: 10.3934/dcds.2013.33.3753

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

Kengo Matsumoto^1,

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

Received: April 30, 2012

Revised: June 30, 2012

Published: July 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 37B10; Secondary: 46L35.

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