We prove that the word problem is undecidable in functionally recursive groups, and that the order problem is undecidable in automata groups, even under the assumption that they are contracting.
We prove that all spaces of finite Assouad-Nagata dimension admit a good order for Travelling Salesman Problem, and provide sufficient conditions under which the converse is true. We formulate a conjectural characterisation of spaces of finite AN -dimension, which would yield a gap statement for the efficiency of orders on metric spaces. Under assumption of doubling, we prove a stronger gap phenomenon about all orders on a given metric space.
We prove that the semigroup generated by a reversible Mealy automaton contains a free subsemigroup of rank two if and only if it contains an element of infinite order.Proposition 2.9. Consider S " X ' Y and let the right action of X preserve
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