Is the injectivity of the global function of a cellular automaton in the hyperbolic plane undecidable?

Abstract: In this paper, we look at the following question. We consider cellular automata in the hyperbolic plane and we consider the global function defined on all possible configurations. Is the injectivity of this function undecidable? The problem was answered positively in the case of the Euclidean plane by Jarkko Kari, in 1994. In the present paper, we give a partial answer: when the configurations are restricted to a certain condition, the problem is undecidable.