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.
Main publications:
(With G.J.Martin) 2-generator arithmetic Kleinian groups, J. fur die Reine und Angew. Math 511 (1999) 95 - 117.
(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.
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