No-splitting property and boundaries of random groups

We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on trees. This implies that their Gromov boundaries, defined at density less than 1/2, are Menger curves.Comment: 20 page

Homotopy bases and finite derivation type for Schutzenberger groups of monoids

Given a finitely presented monoid and a homotopy base for the monoid, and given an arbitrary Schutzenberger group of the monoid, the main result of this paper gives a homotopy base, and presentation, for the Schutzenberger group. In the case that the R-class R' of the Schutzenberger group G(H) has only finitely many H-classes, and there is an element s of the multiplicative right pointwise stabilizer of H, such that under the left action of the monoid on its R-classes the intersection of the orbit of the R-class of s with the inverse orbit of R' is finite, then finiteness of the presentation and of the homotopy base is preserved.Comment: 24 page

Relative rewriting systems

No abstract available

Properties of certain semigroups and their potential as platforms for cryptosystems

In this paper, we study some properties of semigroups with presentation aOE (c) a,b ; a (p) =b (r) ,a (q) =b (s) >. We will also study their potential as platforms for the Diffie-Hellman key exchange protocol