Alex Elzenaar

Publications and preprints

Full list with abstracts and detailed citations: PDF

Preprints

From disc patterns in the plane to character varieties of knot groups, 2025.: arXiv:2503.13829 [math.GT].
Changing topological type of compression bodies through cone manifolds, 2024.: arXiv:2411.17940 [math.GT].
On thin Heckoid and generalised triangle groups in \( \mathrm{PSL}(2,\mathbb{C}) \) (with Gaven Martin and Jeroen Schillewaert), 2024.: arXiv:2409.04438 [math.GR].
Bounding deformation spaces of 2-generator Kleinian groups (with Jianhua Gong, Gaven Martin, and Jeroen Schillewaert), 2024.: arXiv:2405.15970 [math.CV].

Published

Putatively optimal projective spherical designs with little apparent symmetry (with Shayne Waldron), 2025.: Journal of Combinatorial Designs 33; dataset on Zenodo.
Preprint: arXiv:2405.19353 [math.CO], 2024.
Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds (with Gaven Martin and Jeroen Schillewaert), 2024.: 2021–22 MATRIX Annals. MathSciNet: MR4807267.
Preprint: arXiv:2204.11422 [math.GT], 2022. Corrected preprint: PDF.
The combinatorics of the Farey words and their traces (with Gaven Martin and Jeroen Schillewaert), 2024.: Groups, Geometry, and Dynamics, published online first.
Preprint: arXiv:2204.08076 [math.GT], 2022.
Approximations of the Riley slice (with Gaven Martin and Jeroen Schillewaert), 2023.: Expositiones Mathematicae 41. MathSciNet: MR4557273.
Preprint: arXiv:2111.03230 [math.GT], 2021. Corrected preprint: PDF.

Miscellaneous unpublished expository writing

The lake where they had hidden the reflections: Quasi-Fuchsian groups and their embeddings [This was a talk in a reading group.]
Knot knotes
Apocrypha and ephemera on the boundaries of moduli space [More detailed notes on deformations of geometrically finite Kleinian groups are under preparation.]
Uniformisation, equivariance, and vanishing: three kinds of functions hanging around your Riemann surface

Selected talks

Deformations of 3-orbifold holonomy groups and applications, 11 Dec. 2024 (slides): Uni. Auckland: Joint Meeting of the NZMS, AustMS and AMS, Early Career Showcase in Low-Dimensional Topology session.
Limit sets of cone manifolds, 10 Dec. 2024 (poster): Uni. Auckland: Joint Meeting of the NZMS, AustMS and AMS, poster session.
Combinatorial structures in trace polynomials of function groups, 28 Nov. 2024 (slides): ANU: 8th Australian Algebra Conference.
Two-bridge knots, genus two surfaces, and discrete groups with two generators, 12 Nov. 2024 (slides): Uni. Melbourne: Geometry and topology in low dimensions: a conference celebrating the mathematics of Craig Hodgson.
Is \( \mathrm{PSL}(2,\mathbb{Z}) \) discrete?, 24 Jul. 2024 (slides): Monash Uni.: Topology Seminar.
The dynamic in the static: Manifolds, braids, and classical number theory, 10 May 2023 (slides): Uni. Leipzig: Regiomontanus PhD Seminar.
What is a Kleinian group?, 21 Sep. 2022 (slides, recording): Online: Australian Postgraduate Algebra Colloquium.
Pictures of hyperbolic spaces, 4 May 2022 (slides): TU Berlin: Discrete Mathematics and Geometry Seminar.
The Farey polynomials, 6 Dec. 2021 (slides): Uni. Auckland: Waiheke Groups and Geometry Retreat.
Approximating the Riley slice exterior, 2 Dec. 2021 (slides, recording): MATRIX Online: Workshop on Groups and Geometries.
Real varieties of spherical designs, 1 Apr. 2021 (slides): Uni. Auckland: Algebra and Combinatorics Seminar.

Click for a list of other talks, lecture notes, and other ephemera.

7 May 2025: All arithmetic 2-bridge link groups in Topology Group Meeting, slides, notes.
12 March 2025: Rigid systems of poles and hinges in Topology Group Meeting, notes.
Joint Meeting schedule
November 2024: Variations on a theme of Wielenberg.
July 2024: Reading group Hyperbolic Knot Theory, miscellaneous notes.
17, 18 July 2024: Guest lectures for MATHS 782 "Geometric Group Theory" (Uni. of Auckland), lecture notes for lecture 1b (surfaces), lecture notes for lecture 2a (conformal maps), miscellaneous problems, some books
July-August 2023: Minicourse on knot theory and geometry (Uni. of Auckland), below.
16 March 2023: Connectedness of the Hilbert scheme in reading group of Javier and Angel, lecture notes.
17 to 20 January 2023: Apocrypha and ephemera on the boundaries of moduli space, a minicourse at the Uni. of Auckland (also 7 December 2022 at MPI), lecture notes.
10 October 2022: Uniformisation, equivariance, and vanishing: three kinds of functions hanging around your Riemann surface, at MPI, lecture notes.
3 August 2022: Reproducibility in Computer Algebra (MPI MIS), handouts for practical activity (event co-organised with Christiane Görgen and Lars Kastner).
15 July 2022: On the MathRepo page "Farey Polynomials", in the MathRepo: Data for and from your Research event (MPI MIS), slides.
24 May 2022: Projective varieties over \(\mathbb{C}\), in the Lorentzian polynomials day which I organised, slides.
27 April 2022: Strange circles: The Riley slice of quasi-Fuchsian space, in the Seminar on Nonlinear Algebra (MPI MIS), slides.
17 March 2022: Strange circles: The Riley slice of quasi-Fuchsian space, in Pedram Hekmati's seminar on moduli spaces (Uni. of Auckland), slides.
19 July 2021: The moulding of hyperbolic clay: Deformation spaces of Kleinian groups (Uni. of Auckland), presentation slides.
8 June 2021: Some properties of \(2 \times 2 \) matrices, in the UoA Dept. of Mathematics Student Research Conference, extended abstract, presentation slides.
[This talk was one of the four winners of the Best Talk award, along with the talks of Isabelle Steinmann, Chris Pirie, and David Groothuizen Dijkema.]
Very rough lecture notes for the graduate seminar I taught on Kleinian groups in Semester 1, 2021: PDF 1, PDF 2, further reading list, and post-mortem.
October 2020: Tropical geometry and buildings (three lectures), notes.
23 July 2020: Toric varieties (Uni. of Auckland), presentation slides.

Deformation spaces of rank two Kleinian groups

The dual complex to the Farey triangulation.

A rank two Kleinian group is a discrete subgroup of \( \mathrm{PSL}(2,\mathbb{C}) \) generated by two elements. If the group is non-elementary, then it is related in complicated and interesting ways to hyperbolic 3-orbifolds that have boundary at infinity consisting of a genus two Riemann surface.

A graph curve is an algebraic curve consisting of a number of thrice-marked spheres, each marked point corresponding to a node (a transverse intersection of two components). Each graph curve of genus two comes from a trivalent graph on two vertices and three edges. There are exactly two such graphs: the theta graph (each edge joins both nodes), and the handcuff graph (one edge joins the nodes, and the others begin and end on the same node). Both of these graphs are homotopy retracts of the genus two handlebody. Therefore there are only two graph curves of genus two.

On the edge of the deformation space of 3-manifolds with genus two surface at infinity there lie manifolds with the same configuration of spheres at infinity: pairs of thrice-punctured spheres with rank one cusps, with incidence graph a trivalent graph with two vertices and three edges (the incidence graph has vertices correspoding to topological components of the surface and edges corresponding to nodes).

By Thurston's ending lamination theorem (proved for this special case by Minsky and Miyachi), on the boundary of the 3-manifold space you get a different limit for each choice of embedding of the trivalent graph into the handlebody, and you can also take limits of such choices to get 'degenerate' orbifolds—the graphs might even be knotted! Conversely every boundary point arises in this way. So there is a very complicated map from the space of these boundary groups (which is basically a Teichmüller space, up to a small quotient) to the space of graph curves (which has two points). It turns out that this complicated map is basically reflecting the geometry of two-bridge links. Manifolds on the boundary that correspond to handcuff graphs arise from two-bridge links with two components, and manifolds corresponding to theta graphs arise from two-bridge knots. The knots do not live inside the deformation spaces, but they lie on tendrils of discrete groups that creep out beyond the moduli spaces.

The Riley slice is the space of Kleinian groups generated by two parabolic elements such that the quotient manifold is a Conway ball: a 3-ball with two arcs drilled out. Choosing a way of arranging these arcs into a rational tangle is equivalent to picking a simple closed curve on the boundary sphere; suppose that this curve is represented by a hyperbolic element \( W_{p/q} \) with trace \( \mathrm{tr}\, W_{p/q} < -2 \) in the holonomy group of the manifold (actually, you need to pick the correct component of the set of points where this word is hyperbolic, but this is immaterial for the time being). The boundary of the deformation space can be reached by smoothly deforming \( W_{p/q} \) until it is parabolic (trace equals \( -2 \)). Keep deforming \( W_{p/q} \) so that its trace decreases; the group is no longer discrete except sporadically, and these discrete groups correspond to replacing the parabolic arc with a cone arc. Eventually you reach \( \mathrm{tr}\, W_{p/q} = 2\), and in fact \( W_{p/q} = 1 \). You have now reached the fundamental group of the \( p/q \) 2-bridge link. The arc (which has now vanished to become a solid part of the knot complement) is an upper or lower unknotting tunnel for the knot; and the point on the boundary of the deformation space where this arc was parabolic corresponds to the manifold where both the knot and the unknotting tunnel have been drilled out as parabolic arcs from \( \mathbb{S}^3 \).

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).
Various authors have written papers in different areas. Some of the most important to us include The Riley slice of Schottky space (Keen and Series), Parabolic representations of knot groups (Riley), Classification of non-free Kleinian groups generated by two parabolic transformations (Akiyoshi, Ohshika, Parker, Sakuma and Yoshida), Cusps in complex boundaries of one-dimensional Teichmüller space (Miyachi), and The tree of knot tunnels (Cho and McCullough).
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).

Cone manifolds and discrete geometry of indiscrete groups

An indiscrete deformation of a genus two function group with controlled cone angle.

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.

Poster: Limit sets of cone manifolds; preprint: Changing topological type of compression bodies through cone manifolds.

More to come in 2025.

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.

9.30am—Matroids and chromatic polynomials (Tobias Boege, MPI MiS): Slides
10:15am—Varieties over C and embeddings into projective space via elliptic curves (Lukas Zobernig, The University of Auckland): Slides
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).

1:30pm—Convex geometry & mixed volumes (Mara Belotti, TU Berlin): Slides
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.