We study the group of orientation-preserving homeomorphisms of the real line R which are piecewise-linear with respect to a finite subdivision of R; we denote this group PLF(R). Our main results are presentations of PLF(~) and certain of its subgroups; a proof that PLF(R) contains no free subgroups of rank greater than 1; and a proof that PLF(R) satisfies no laws. In the process, we construct a sequence of finitely-presented subgroups G(p) of PLF(R) which contain no free subgroups of rank greater than 1 and satisfy no laws. We also briefly consider the full group of piecewise-linear homeomorphisms of R and the group of piecewise-linear homeomorphisms of the circle S 1.
Abstract. A slight modification of the Burau representation of the braid group is shown to be unitary relative to an explicitly defined Hermitian form. This gives a partial answer to the problem of identifying the image of the Burau representation and provides a tool for attacking the question of whether or not the Burau representation is faithful.In [2, problem 14, p. An important open question is whether or not the Burau representation is faithful. This is known for n < 3 [4] (also see [2]). It seems likely that the results presented here will help answer this question for n > 4.In §1 we establish some conventions about matrices and in §2 we describe the Burau representation. In §3 we prove the theorem and justify the remarks following its statement above. Finally, in §4 we offer two conjectures which, taken together, would imply that the Burau representation is faithful.1. Preliminaries. Let Z denote the ring of (rational) integers, let t be an indeterminate and let Z[t, /"'] denote the ring of Laurent polynomials with coefficients in Z. We consider infinite-dimensional matrices A = (a¡ ) with entries in Z [f, /"']
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