Growth in Topology and Number Theory: Volumes, Entropy, and L2-torsion

Tuesday, July 10

09:30 -10:20 Gerrit Herrmann: Thurston norm and L^2-Betti numbers
10:30 - 11:00 Group photo and coffee break
11:00 - 11:50 Abhijit Champanerkar: Geometry of biperiodic alternating links
12:00 - 14:00 Lunch Break
14:00 - 14:50 Viveka Erlandsson: Counting curves on surfaces
15:00 - 15:30 Coffee Break
15:30 - 16:20 Alan Reid: Confinal sequences of lattices containing surface subgroups of bounded genus
16:40 - 17:30 Stephan Tillmann: Thurston norm via spun-normal surfaces
18:30 Conference dinner at "Roses" (roses-bonn.de)

Wednesday, July 11

09:30 -10:20 Jean Raimbault: Finiteness results for topological invariants of arithmetic 3—manifolds
10:30 - 11:00 Coffee Break
11:00 - 11:50 Vincent Emery: Hyperbolic manifolds and pseudo-arithmeticity
after 11:50 Free times for having lunch
13:00 We offer an excursion to the Drachenfels (meeting point at Endenicher Allee 60)

Friday, July 13

09:30 -10:20 Yohei Komori: Growth of hyperbolic Coxeter groups
10:30 - 11:00 Coffee Break
11:00 - 11:50 Christopher Smyth: Closed sets of Mahler measures
12:10 - 13:00 Final discussion / Problem session

 End of the conference

Abstracts

Abhijit Champanerkar: Geometry of biperiodic alternating links

In this talk we discuss the hyperbolic geometry of alternating link complements in the thickened torus. We show that many such links are hyperbolic and admit a positively oriented, unimodular geometric triangulation. For these links, we also determine sharp upper and lower volume bounds. For semi-regular links, which are links that project to semi-regular Euclidean tilings, we determine their complete hyperbolic structure. This has several nice consequences like determining exact volumes, arithmeticity and commensurability for this class of links. We will discuss relations to the asymptotic geometry of hyperbolic links in the 3-sphere which approach biperiodic links. This is joint work with Ilya Kofman and Jessica Purcell.

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Nathan M. Dunfield: Building hyperbolic 3-manifolds for fun and profit

I will construct families of hyperbolic 3-manifolds that exhibit different behaviors relevant to understanding torsion growth in hyperbolic 3-manifolds. For example, I will show how to build hyperbolic 3-manifolds with injectivity radius bounded below whose "homological regulator" is exponentially large in the volume. I will also discuss examples probing how the injectivity radius interacts with the relationship between the harmonic and Thurston norms on the first cohomology. Finally, I will discuss examples where I try to control the "relative Cheeger constant" of Lipnowski-Stern that they related to the first eigenvalue of the Laplacian on 1-forms. This is mostly joint work with Jeff Brock.

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Vincent Emery: Hyperbolic manifolds and pseudo-arithmeticity

I will introduce and motivate a notion of pseudo-arithmeticity, which possibly applies to all lattices in PO(n,1) with n>3. This is joint work with Olivier Mila (Bern).

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Viveka Erlandsson: Counting curves on surfaces

Two curves in a closed hyperbolic surface of genus g are of the same type if they differ by a mapping class. Mirzakhani studied the number of curves of given type and of hyperbolic length bounded by L, showing that as L grows, it is asymptotic to a constant times L^{6g-6}. In this talk I will discuss generalizations and applications of this result. In particular we will see that the same asymptotic growth holds for when length is induced by any Riemannian metric. The main ingredient in this generalization is to study measures on the space of geodesic currents.

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Gerrit Herrmann: Thurston norm and L^2-Betti numbers

The Thurston norm measures the minimal complexity of surfaces in 3-manifolds. We will show that relative L^2-Betti numbers detect whether a given surface is Thurston norm minimizing.

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Eriko Hironaka: Dilatations of pseudo-Anosov mapping classes

In this talk we discuss some recent progress on the minimum dilatation problem for pseudo-Anosov mapping classes.  As a key tool, we will develop and apply the notion of directed train tracks for pseudo-Anosov suspension flows.

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Andrei Jaikin: Recognition of fibring in compact 3-manifolds

Let M be a compact orientable 3-manifold. We show that if the profinite completion of  pi_1(M) is isomorphic to the profinite completion of a free-by-cyclic group or to the profinite completion of a surface-by-cyclic group, then M fibres over the circle with compact fibre.

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Holger Kammeyer: Profiniteness questions for l²-Betti numbers of S-arithmetic groups

While the first l²-Betti number of a finitely presented residually finite group is determined by the profinite completion, we give examples of S-arithmetic groups showing that this is not true for any higher l²-Betti number. However, we show profiniteness results for higher l²-Betti numbers of S-arithmetic groups in a more restrictive setting. In particular, the sign of the Euler characteristic of arithmetic groups is profinite. Joint work with S. Kionke, J. Raimbault, and R. Sauer.

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Ruth Kellerhals: Higher dimensional modular groups and their geometry

I shall present some new results about quaternionic modular groups, their fundamental polyhedral and the determination of their exact volumes. Of importance is the geometric description in terms of ideal hyperbolic k-rectified regular polyhedra and Napier cycles.

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Yohei Komori: Growth of hyperbolic Coxeter groups

In this talk I will give an overview of recent progress on arithmetic aspects of growth related to hyperbolic Coxeter groups.

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Clara Löh: A dynamical view on simplicial volume

Simplicial volumes count the number of singular simplices needed to reconstruct manifolds; depending on the chosen coefficients, these invariants have different flavours (e.g., geometric, combinatorial, dynamical). In this talk, I will survey the dynamical view on simplicial volumes and how this can help to understand integral simplicial volume of finite coverings and the relation with $L^2$-invariants.

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Bruno Martelli: Hyperbolic Dehn filling in dimension four

Thurston's hyperbolic Dehn filling is a fundamental tool in dimension three that allows us to deform a cusped hyperbolic 3-manifold along a path of hyperbolic cone-manifolds, until it transforms into another hyperbolic 3-manifold. We show that this tool is sometimes available also in dimension four. The only examples we know so far are constructed using deforming families of Coxeter polytopes.
(Joint with Stefano Riolo.)

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Jean Raimbault: Finiteness results for topological invariants of arithmetic 3—manifolds

Recent work of M. Fraczyk implies that only finitely many arithmetic hyperbolic 3--manifolds "of congruence type" can have a given Heegard or fibration genus. I will discuss these and similar results for other invariants, in particular the type of manifolds one can obtain by Dehn surgery on a manifold with cusps.

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Alan Reid: Confinal sequences of lattices containing surface subgroups of bounded genus

In this talk we describe a construction of non-uniform lattices G in SL(3,R) that have the property that there is a sequence of finite index normal subgroups {G_i} which intersect in the identity and all G_i contain a surface subgroup isomorphic to a genus 3 surface group. This contrasts sharply with lattices in rank 1 Lie groups.

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Roman Sauer: Riemannian volume vs simplicial volume and l2-Betti numbers

We discuss recent and less recent developments around the relationship between l2-Betti numbers and volume on aspherical Riemannian manifolds. An important tool in this discussion are random coverings.

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Christopher Smyth: Closed sets of Mahler measures

In 1981 David Boyd conjectured that the set of all Mahler measures of polynomials with integer coefficients, in any number of variables, is closed. I report on some progress towards a proof of this conjecture. I also describe a possible explanation of how the smallest non-trivial Mahler measures arise.

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Stephan Tillmann: Thurston norm via spun-normal surfaces

Let M be the complement of a link with at least two components in a homology 3-sphere. I will describe an algorithm to compute the unit ball of the Thurston norm on second homology of M using spun-normal surfaces in an ideal triangulation of M. Applications, experimental results, and extensions will be discussed. This is joint work with Daryl Cooper and William Worden.

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Giulio Tiozzo: An introduction to core entropy

The notion of topological entropy, arising from information theory, is a fundamental tool to understand the complexity of a dynamical system. When the dynamical system varies in a family, the natural question arises of how the entropy changes with the parameter.

In the last decade, W. Thurston introduced these ideas in the context of complex dynamics by defining the "core entropy" of a quadratic polynomials as the entropy of a certain forward-invariant set of the Julia set (the Hubbard tree).

As we shall see, the core entropy is a purely topological / combinatorial quantity which nonetheless captures the richness of the fractal structure of the Mandelbrot set. In particular, we will relate the variation of such a function to the geometry of the Mandelbrot set. We will also prove that the core entropy on the space of polynomials of a given degree varies continuously, answering a question of Thurston.

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