Complex hyperbolic triangle groups

The theory of complex hyperbolic discrete groups is still in its childhood but promises to grow into a rich subfield of geometry. In this paper I will discuss some recent progress that has been made on complex hyperbolic deformations of the modular group and, more generally, triangle groups. These are some of the simplest nontrivial complex hyperbolic discrete groups. In particular, I will talk about my recent discovery of a closed real hyperbolic 3-manifold which appears as the manifold at infinity for a complex hyperbolic discrete group

Traces in Complex Hyperbolic Triangle Groups

We present several formulas for the traces of elements in complex hyperbolic triangle groups generated by complex reflections. The space of such groups of fixed signature is of real dimension one. We parameterise this space by a real invariant alpha of triangles in the complex hyperbolic plane. The main result of the paper is a formula, which expresses the trace of an element of the group as a Laurent polynomial in exp(i alpha) with coefficients independent of alpha and computable using a certain combinatorial winding number. We also give a recursion formula for these Laurent polynomials and generalise the trace formulas for the groups generated by complex mu-reflections. We apply these formulas to prove some discreteness and some non-discreteness results for complex hyperbolic triangle groups.Comment: 22 pages, 1 figure; prop. 11 added, typos corrected; cor. 19 removed (not correct

A better proof of the Goldman-Parker conjecture

The Goldman-Parker Conjecture classifies the complex hyperbolic C-reflection ideal triangle groups up to discreteness. We proved the Goldman-Parker Conjecture in [Ann. of Math. 153 (2001) 533--598] using a rigorous computer-assisted proof. In this paper we give a new and improved proof of the Goldman-Parker Conjecture. While the proof relies on the computer for extensive guidance, the proof itself is traditional.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper35.abs.htm

Recent Progress in the Symmetric Generation of Groups

Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for groups and practically by providing succinct means of representing group elements. We give a survey of results obtained in the study of these symmetric generating sets. In keeping with earlier surveys on this matter, we emphasize the sporadic simple groups. ADDENDUM: This is an updated version of a survey article originally accepted for inclusion in the proceedings of the 2009 `Groups St Andrews' conference. Since the article was accepted the author has become aware of other recent work in the subject that we incorporate to provide an updated version here (the most notable addition being the contents of Section 3.4.)Comment: 14 pages, 1 figure, an updated version of a survey article accepted for the proceedings of the 2009 "Groups St Andrews" conference. v2 adds McLaughlin reference and abelian groups reference

The Asymptotic distribution of circles in the orbits of Kleinian groups

Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either the critical exponent of Gamma is strictly bigger than 1 or P does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Gamma. Our construction also works for P invariant under a geometrically infinite group Gamma, provided Gamma admits a finite Bowen-Margulis-Sullivan measure and the Gamma-skinning size of P is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.Comment: 31 pages, 8 figures. Final version. To appear in Inventiones Mat

Groups generated by two elliptic elements in PU(2,1)

Let $f$ and $g$ be two elliptic elements in $\mathbf{PU}(2,1)$ of order $m$ and $n$ respectively, where $m\geq n>2$. We prove that if the distance $\delta(f,g)$ between the complex lines or points fixed by $f$ and $g$ is large than a certain number, then the group $$ is discrete nonelementary and isomorphic to the free product $\mathbf{Z}_{m}*\mathbf{Z}_{n}$.Comment: 9 page