Alex Elzenaar

Peripheral subgroups of Kleinian groups

Here is the preprint, and here is how I explained it to a friend of mine:

You have a group \(G\), and you know it is generated by some matrices in \( \mathrm{PSL}(2,\mathbb{C}) \). You also know that \( \mathbb{H}^3/G \) is a nice enough manifold ("geometrically finite"). If you wiggle the entries of the generators, the result is a group \(G'\) with the property that if you squint, \( \mathbb{H}^3/G \)' looks the same as \( \mathbb{H}^3/G \). (The technical term is, they are quasi-isometric.) I can say with polynomial inequalities over the field \(\mathbb{R}\) (i.e. semialgebraically) how far you can wiggle \(G'\) before it stops looking like \(G\) when you squint. This is Main Theorem B. The way I do this is by finding the polygon with which \(G\) tiles the Riemann sphere—this is Main Theorem A—and then defining a way to wiggle the polygon along with the matrices, so that as you get a new group G' you also get a new polygon. The semi-algebraic sets are the sets that detect when the polygon collides with itself & stops tiling the Riemann sphere. This is detecting "peripheral structures" because the Riemann sphere is the boundary of \(\mathbb{H}^3\), so I am studying what is happening "at the periphery" of the manifold as you wiggle the metric. This is important because I can cover the entire set of possible metrics for the topological manifold \( \mathbb{H}^3/G \) with these sets. If you take a random group now you can check whether it lies in any of the sets, and if it does then you know the group is discrete. This is interesting since discreteness checking is a very hard open problem (Section 1.3).

The proofs involve careful analysis of fundamental domains of Kleinian groups using hyperbolic geometry of the ends of their quotient manifolds. These domains are generalised Ford domains, in the sense that their sides are circles which lie in the pencils determined by the isometric circles of elements of the group; but we choose special elements which are `visible from the periphery' instead of taking all elements in the group like with a normal Ford domain. The deformations of these domains are studied with techniques from real algebraic geometry and Möbius geometry. See also talks at the Joint Meeting of the NZMS, AustMS, and AMS and the 8th Australian Algebra Conference. Note, both those talks were on preliminary versions of this work and do not entirely capture the spirit of the final results.

Cone manifolds and discrete geometry of indiscrete groups

The abstract I submitted for my poster at the joint meeting of the NZMS, AustMS and AMS is a good introduction to what I am thinking about currently:

Tilings of the plane or of more general 2D geometries are very classical and central mathematical objects. Their symmetry groups are discrete, and the quotients of the tilings by their symmetry groups are in the best case smooth surfaces and in the worst case surfaces with some bits that look like paper cones: take a disc of paper, cut a triangle out, and glue the resulting sides together. If you start with a tiling of the Euclidean or hyperbolic plane, the angles must be submultiples of \( \pi \). But if you're cutting triangles out of a piece of paper, there is no physical reason that you can't pick any angle you want: you still get a cone. Unwrap the corresponding surface, and you get a subset of the plane: it's just that the symmetry group is no longer discrete and if you try to make a tiling from it everything will overlap. Lots of abstract geometric group theory fails in this new setting, but if all you care about is taking a piece of paper, cutting angles out, and gluing a bunch of copies of the results together regardless, how much theory can you recover? We present some computer experiments and preliminary results in 3D: instead of gluing sheets of paper with corners, we glue imaginary blocks of hyperbolic space with corners.

Gluing 2-handles into compression bodies

• Preprint: Changing topological type of compression bodies through cone manifolds.
• Poster: Limit sets of cone manifolds.
• Animation: compression body cone manifolds.
• Talk slides: October 2025

Drilling tunnels in knots and links

• Preprint: Expansion joints in hyperbolic manifolds.
• Animation: Borromean rings to lantern manifold.

Deformation spaces of rank two Kleinian groups

The Riley slice is the space of Kleinian groups generated by two parabolic elements such that the quotient manifold is a 3-ball with a rational tangle drilled out. Choosing the rational tangle is equivalent to picking a simple closed curve on the boundary sphere. This curve is represented by a word \( W_{p/q} \) in the rank two free group. There exists a smooth family of groups generated by two parabolics parameterised by \( \mathrm{tr}\, W_{p/q} \in (-\infty,+2] \). When the trace is less than \( -2 \), the corresponding group lies in the Riley slice. When the trace equals \( -2 \), the surface degenerates to a pair of thrice-punctured spheres and the group is on the boundary of the Riley slice. Between \( -2 \) and \( 2 \), the group is only sporadically discrete. Eventually when \( \mathrm{tr}\, W_{p/q} = 2\) the word \( W_{p/q} \) becomes the identity and the group is the fundamental group of the \( p/q \) 2-bridge link.

If this sounds interesting:

• You might want to start with our expository article Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds (joint work with Gaven Martin and Jeroen Schillewaert) which we wrote to give historical and mathematical background: we aimed for this to be accessible to beginning graduate students with only a little complex analysis and topology knowledge.
• (To appear) A comprehensive study of the groups generated by pairs of parabolic and elliptic elements, following work of Keen and Series and various others (joint with Martin and Schillewaert).
• Work related to the discreteness problem: Approximations of the Riley slice (joint with Martin and Schillewaert); Bounding deformation spaces of 2-generator Kleinian groups (joint with Gong, Martin, and Schillewaert)
• A detailed abstract combinatorial study of certain polynomials which control the geometry of these groups: The combinatorics of Farey words and their traces (joint with Martin and Schillewaert), table to denominator 10.
• Applications to the enumeration of arithmetic groups: On Thin Heckoid and Generalised Triangle Groups in \( \mathrm{PSL}(2,\mathbb{C}) \) (joint with Martin and Schillewaert).

A zoo of Kleinian groups

There are several useful 'zoos' of Kleinian groups with interesting properties; I collected several interesting groups and families of groups from a few sources, and you can find their limit sets on this page.

These images were produced using the Bella computational package for Kleinian groups. A previous version of this package: Riley slice computational package (GitHub). With this earlier package I produced some animations, and some more limit sets. Some more visually impressive animations can be found on the website of Emily Dumas.

Minicourse on knot theory and geometry

In July 2023 I organised a minicourse on knot theory at the University of Auckland, focusing on the representation theory of holonomy groups. View the abstract or download the latest version of the notes.

Erratum: the Seifert surface of the figure eight knot drawn in the notes is not correct. A correct application of Seifert's algorithm and associated sketch may be found in L. Kauffman, On knots (Princeton), cited at that point in the notes.

There will be eight lectures over four weeks in 303.148 (for the first two weeks at least):

	Wed, 2pm	Fri, 2pm
Classical knot theory	5 Jul: Basics	7 Jul: Fundamental group
Geometric knot theory	12 Jul: Knot complements	14 Jul: Hyperbolic invariants
Braids	19 Jul: Two-bridge knots	21 Jul: Braids and mapping class groups
Knot polynomials	26 Jul: Classical	28 Jul: Quantum

Josh Lehman gave the lecture on mapping class groups and Lavender Marshall gave the lecture on the Alexander polynomial.

Some useful links:

• From lecture 1.1: Conway: Knots Don't Cancel
• Hyperbolic geometry background for week 2: an intro, VR
• From lecture 2.1: The Geometries of 3 Manifolds, Belt trick, (2,3) Seifert fibration
• Problem session 1 (17 July): problems
• Problem session 2 (31 July): problems

Lorentzian polynomials and algebraic geometry on matroids

If \( X \) is a sufficiently nice variety, the Chow group \( A^*(X) \) provides a homology theory on \( X \); in fact, it admits a ring structure coming from the intersection product. It turns out that such a theory can be made to work on more general spaces, for example one can define a Chow ring for matroids; then the various Hodge-type results (Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations) carry over. Various nice polynomials can be defined with respect to this generalised Hodge theory and the associated cones of 'ample divisors' (which turn out to be submodular functions); these are the Lorentzian polynomials of Brändén and Huh.

A Day of Geometry and Lorentzian Polynomials

At the end of May 2022 there was a seminar at the Institut Mittag-Leffler on the work of Branden, Huh, Katz, and various other people on Lorentzian polynomials and the geometry of matroids; before this event on Tuesday 24 May, I organised a very informal Zoom workshop on some of the geometric background material.

Abstract. Even if you do not know what Lorentzian polynomials are, you may have heard of Minkowski volume polynomials, the polynomials of the form \( \mathrm{vol}(x_1 K_1 + \cdots + x_n K_n) \) where \( K_1,\ldots,K_n \) are convex bodies—and these are somehow the "canonical examples" of Lorentzian polynomials. The goal of the workshop is to give many different examples of Lorentzian polynomials arising in geometry. The talks will be very informal, non-technical, and have many pictures.

The final schedule was as follows (all times are CET). Many of the speakers have kindly allowed me to share their slides and/or lecture notes.

1. 9.30am—Matroids and chromatic polynomials (Tobias Boege, MPI MiS): Slides
2. 10:15am—Varieties over C and embeddings into projective space via elliptic curves (Lukas Zobernig, The University of Auckland): Slides
3. 11:00am—Hyperbolic polynomials (Hisha Nguyen, V.N. Karazin Kharkiv National University)

• Break (hopefully the morning talks are finished by 11:45, or 12 at the latest if we run over time).

4. 1:30pm—Convex geometry & mixed volumes (Mara Belotti, TU Berlin): Slides
5. 2:15pm—Projective varieties over \( \mathbb{C} \) (Alex Elzenaar, MPI MiS): Slides

Some background material

• Petter Brändén, Jonathan Leake: Lorentzian polynomials on cones and the Heron-Rota-Welsh conjecture
• Petter Brändén, June Huh: Lorentzian polynomials (direct multivariable-calculus proof of the theory without mentioning Hodge theory). See also this lecture by June Huh
• Matthew Baker: Hodge theory in combinatorics (an expository paper)
• Karim Adiprasito, June Huh, Eric Katz: Hodge Theory for Combinatorial Geometries (definition of the Hodge ring for arbitrary matroids)

Spherical designs

A spherical \((3,3)\)-design in \( \mathbb{R}^3 \) of 16 vectors. Spherical \((t,t)\)-designs are arrangements of points on the sphere (possibly with weights) which are spaced 'far apart from each other': they are finite sets in \( \mathbb{R}^d \) such that the integral over the sphere of each homogeneous polynomial of degree \(2t\) in \( d \) variables is equal to its average value on the set. There are generalisations of this definition to subsets of \( \mathbb{C}^d \) and \( \mathbb{H}^d \) (the \(d\)-fold product of the Hamiltonian quaternion algebra, not hyperbolic \(d\)-space!).

Optimal designs and near-designs

Shayne Waldron and I have a paper in preparation: Putatively optimal projective spherical designs with little apparent symmetry, computing various spherical designs in order to find those of minimal order; a large set of designs and near-designs are archived on on Zenodo at DOI: 10.5281/zenodo.6443356. You can look at the code used to generate these on GitHub; it uses the Manopt optimisation toolbox. This work was was funded in part by a University of Auckland Summer Research Scholarship (2019-20). You can view the final report for the scholarship.

Spherical designs and sums of squares

• Talk at the UoA algebra and combinatorics seminar, April 2021: Presentation slides.

BSc(hons) dissertation and MSc thesis

I completed my BSc(Hons) dissertation in 2020 under the supervision of Dr. Jeroen Schillewaert.

My Master of Science thesis was completed in 2021-22 in the Department of Mathematics at the University of Auckland, under the supervision of Dist. Prof. Gaven Martin (NZ Institute of Advanced Study, Massey University) and Dr. Jeroen Schillewaert. For more information see the section on deformation spaces above.