Dr Maclachlan's main interests are related to discrete groups of isometries of hyperbolic space. This seemingly rather specialised area has non-trivial intersection with a number of other topics which are being actively investigated and in which interesting problems arise. These include: very low-dimensional topology and geometry, particularly of 3-manifolds; group theory, particularly combinatorial group theory arising from the fundamental groups of orbifolds and finite groups arising as automorphism groups of manifolds; number theory in the guise of arithmetic Kleinian and Fuchsian groups and particularly the impact of and, and input from, number theoretic techniques on the geometry of hyperbolic manifolds.

• (With G.J.Martin) 2-generator arithmetic Kleinian groups, J. fur die Reine und Angew. Math 511 (1999) 95 - 117.
• (With G.J. Martin) The non-compact arithmetic generalised triangle groups, Topology 40 (2001) 927 - 944.
• (With M.Conder, S. Todorovic Vasilevic and S. Wilson) Bounds for the number of automorphisms of a compact non-orientable surface, J. London Math. Soc. 68 (2003) 65 - 82.
• (With A.W. Reid) The Arithmetic of Hyperbolic 3-manifolds, Springer Graduate Texts in Maths. Vol 219 (2003)
• (With M. Conder) Compact hyperbolic 4-manifolds of small volume, Proc. American Math. Soc. 133 (2005) 2469 - 2476.
• Torsion in arithmetic Fuchsian groups, J. London Math. Soc. 73 (2006) 14 - 30
• (With A.W. Reid and D.D. Long) Arithmetic Fuchsian groups of genus 0, Pure and Applied Math. Quarterly (for J.H. Coates) 2 (2006) 569 - 599.

• Institute of Mathematics University of Aberdeen