Using HNN extensions of the Boone-Britton group, a group E is obtained which simulates Turing machine computation in linear space and cubic time. Space in E is measured by the length of words, and time by the number of substitutions of defining relators and conjugations by generators required to convert one word to another. The space bound is used to derive a PSPACE-complete problem for a topological model of computation previously used to characterize NP-completeness and RE-completeness.