Dense 2-generator subsemigroups of 2 x 2 matrices

We construct explicit examples of 2-generator dense subsemigroups of CL(2, R) and SL(2, R). In the complex case, we prove the existence of uncountably many 2-generator dense subsemigroups of GL(2, C). We also show that the semigroup of real linear fractional transformations on a proper subinterval of the real line does not admit any 2-generator dense subsemigroups, and then we construct a 3-parameter family of examples of 3-generator dense subsemigroups. (C) 2011 Elsevier Inc. All rights reserved.