This paper looks at the class of groups admitting normal forms for which the right multiplication by a group element is computed in linear time on a multi-tape Turing machine. We show that the groups $\mathbb{Z}_2 \wr \mathbb{Z}^2$, $\mathbb{Z}_2 \wr \mathbb{F}_2$ and Thompson's group $F$ have normal forms for which the right multiplication by a group element is computed in linear time on a $2$-tape Turing machine. This refines the results previously established by Elder and the authors that these groups are Cayley polynomial-time computable.
Comment: Published in journal of Groups, Complexity, Cryptology