High-mobility semiconductor fins are formed on an insulator layer using techniques allowing precise control of fin heights. Lattice-matched fins are grown epitaxially on sidewalls of an essentially defect-free portion of a semiconductor template. The fins are formed within laterally extending trenches in a top dielectric layer, the thickness of which determines fin height. The trenches extend orthogonally to the template. Epitaxial overgrowth above the top dielectric layer is removed by planarization. The fin template and top dielectric layer are removed, leaving sets of parallel fins on the insulator layer. The fin template can be replaced by an isolation region for electrically isolating sets of fins.