# Dawid Kielak

### Tutorial Fellow in Pure Mathematics, Hertford College, Oxford

My research interests lie somewhere between Geometric Group Theory and the theory of group rings. I am a member of the Algebra and Topology research groups. My work is funded by the ERC Starting Grant "Fibring".

I am a member of the LMS and the EMS.

## Preprints and publications

• Profinite rigidity of fibring (with Sam Hughes)

We introduce the classes of TAP groups, in which various types of algebraic fibring are detected by the non-vanishing of twisted Alexander polynomials. We show that finitely presented LERF groups lie in the class $\mathsf{TAP}_1(R)$ for every integral domain $R$, and deduce that algebraic fibring is a profinite property for such groups. We offer stronger results for algebraic fibring of products of limit groups, as well as applications to profinite rigidity of Poincaré duality groups in dimension 3 and RFRS groups.

arXiv
• Origami expanders (with Goulnara Arzhantseva, Tim de Laat, and Damian Sawicki)

We construct a new type of expanders, from measure-preserving affine actions with spectral gap on origami surfaces, in each genus $g\geqslant 1$. These actions are the first examples of actions with spectral gap on surfaces of genus $g>1$. We prove that the new expanders are coarsely distinct from the classical expanders obtained via the Laplacian as Cayley graphs of finite quotients of a group. In genus $g=1$, this implies that the Margulis expander, and hence the Gabber–Galil expander, is coarsely distinct from the Selberg expander. For the proof, we use the concept of piecewise asymptotic dimension and show a coarse rigidity result: A coarse embedding of either $\mathbb R^2$ or $\mathbb H^2$ into either $\mathbb R^2$ or $\mathbb H^2$ is a quasi-isometry.

arXiv
• Agrarian and $\ell^2$-Betti numbers of locally indicable groups, with a twist (with Bin Sun)

We prove that twisted $\ell^2$-Betti numbers of locally indicable groups are equal to the usual $\ell^2$-Betti numbers rescaled by the dimension of the twisting representation; this answers a question of Lück for this class of groups. It also leads to two formulae: given a fibration $E$ with base space $B$ having locally indicable fundamental group, and with a simply-connected fibre $F$, the first formula bounds $\ell^2$-Betti numbers $b^{(2)}_i(E)$ of E in terms of $\ell^2$-Betti numbers of $B$ and usual Betti numbers of $F$; the second formula computes $b^{(2)}_i(E)$ exactly in terms of the same data, provided that $F$ is a high-dimensional sphere.

We also present an inequality between twisted Alexander and Thurston norms for free-by-cyclic groups and $3$-manifolds. The technical tools we use come from the theory of generalised agrarian invariants, whose study we initiate in this paper.

arXiv
• Algebraically hyperbolic groups (with Giles Gardam and Alan Logan)

We initiate the study of torsion-free algebraically hyperbolic groups; these groups generalise, and are intricately related to, groups with no Baumslag-Solitar subgroups. Indeed, for groups of cohomological dimension 2 we prove that algebraic hyperbolicity is equivalent to containing no Baumslag-Solitar subgroups. This links algebraically hyperbolic groups to two famous questions of Gromov; recent work has shown these questions to have negative answers in general, but they remain open for groups of cohomological dimension 2.

We also prove that algebraically hyperbolic groups are CSA, and so have canonical abelian JSJ-decompositions. In the two-generated case we give a precise description of the form of these decompositions.

arXiv
• JSJ decompositions and polytopes for two-generator one-relator groups (with Giles Gardam and Alan Logan)

We provide a direct connection between the $\mathcal{Z}_{\max}$ (or essential) JSJ decomposition and the Friedl--Tillmann polytope of a hyperbolic two-generator one-relator group with abelianisation of rank $2$.

We deduce various structural and algorithmic properties, like the existence of a quadratic-time algorithm computing the $\mathcal{Z}_{\max}$-JSJ decomposition of such groups.

arXiv talk in Newcastle
1. The Surface Group Conjectures for groups with two generators (with Giles Gardam and Alan Logan)

The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.

arXiv • to appear in Math. Res. Lett.
2. $\ell^2$ Betti numbers and coherence of random groups (with Robert Kropholler and Gareth Wilkes)

We study $\ell^2$ Betti numbers, coherence, and virtual fibring of random groups in the few-relator model. In particular, random groups with negative Euler characteristic are coherent, have $\ell^2$ homology concentrated in dimension 1, and embed in a virtually free-by-cyclic group with high probability. Similar results are shown with positive probability in the zero Euler characteristic case.

arXiv J. Lond. Math. Soc. (2022) Oberwolfach Report
3. Handlebody Bundles and Polytopes (with Sebastian Hensel)

We construct examples of fibered three-manifolds with first Betti number at least $2$ and with fibered faces all of whose monodromies extend to a handlebody.

arXiv Algebr. Geom. Topol. (2021)
4. Agrarian and $L^2$-invariants (with Fabian Henneke)

We develop the theory of agrarian invariants, which are algebraic counterparts to $L^2$-invariants. Specifically, we introduce the notions of agrarian Betti numbers, agrarian acyclicity, agrarian torsion and agrarian polytope. We use the agrarian invariants to solve the torsion-free case of a conjecture of Friedl-Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are independent of the presentations used. We also show that, for such groups, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl-Tillmann and Friedl-Lück-Tillmann. Finally, we prove that for agrarian groups of deficiency $1$, the agrarian polytope admits a marking of its vertices which controls the Bieri-Neumann-Strebel invariant of the group, improving a result of the second author and partially answering a question of Friedl-Tillmann.

arXiv Fund. Math. (2021)
5. Isoperimetric inequalities for Poincaré duality groups (with Peter Kropholler)

We show that every oriented $n$-dimensional Poincaré duality group over a $*$-ring $R$ is amenable or satisfies a linear homological isoperimetric inequality in dimension $n-1$. As an application, we prove the Tits alternative for such groups when $n=2$. We then deduce a new proof of the fact that when $n=2$ and $R = \mathbb Z$ then the group in question is a surface group.

arXiv Proc. Amer. Math. Soc. (2021) talk in Warwick and Newcastle
6. The Farrell-Jones Conjecture for normally poly-free groups (with Benjamin Brück and Xiaolei Wu)

We prove the $K$- and $L$-theoretic Farrell-Jones Conjecture with coefficients in an additive category for every normally poly-free group, in particular for even Artin groups of FC-type, and for all groups of the form $A \rtimes \mathbb{Z}$ where $A$ is a right-angled Artin group. Our proof relies on the work of Bestvina-Fujiwara-Wigglesworth on the Farrell-Jones Conjecture for free-by-cyclic groups.

arXiv Proc. Amer. Math. Soc. (2021)
7. On property (T) for $\mathrm{Aut}(F_n)$ and $\mathrm{SL}_n(\mathbb Z)$ (with Marek Kaluba and Piotr Nowak)

We prove that $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for every $n \geqslant 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n\geqslant 5$.

We also provide explicit lower bounds for the Kazhdan constants of $\mathrm{SAut}(F_n)$ (with $n \geqslant 6$) and of $\mathrm{SL}_n(\mathbb Z)$ (with $n \geqslant 3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n> 6$.

arXiv Ann. of Math. (2021) case study article in "Nauka w Polsce" in Polish and English
8. The agrarian polytope of two-generator one-relator groups (with Fabian Henneke)

Relying on the theory of agrarian invariants introduced in previous work, we solve a conjecture of Friedl-Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are independent of the presentations used. We also show that, when the groups are additionally torsion-free, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl-Tillmann and Friedl-Lück-Tillmann.

arXiv J. Lond. Math. Soc. (2020)
9. On the smallest non-trivial quotients of mapping class groups (with Emilio Pierro)

We prove that the smallest non-trivial quotient of the mapping class group of a connected orientable surface of genus at least $3$ without punctures is $\mathrm{Sp}_{2g}(2)$, thus confirming a conjecture of Zimmermann. In the process, we generalise Korkmaz's results on $\mathbb C$-linear representations of mapping class groups to projective representations over any field.

arXiv Groups Geom. Dyn. (2020)
10. Residually finite rationally solvable groups and virtual fibring

We show that a finitely generated residually finite rationally solvable (or RFRS) group $G$ is virtually fibred, in the sense that it admits a virtual surjection to $\mathbb{Z}$ with a finitely generated kernel, if and only if the first $L^2$-Betti number of $G$ vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of $3$-manifolds.

arXiv J. Amer. Math. Soc. (2020) Oberwolfach Report
11. The Bieri-Neumann-Strebel invariants via Newton polytopes

We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri-Neumann-Strebel (BNS) via a theorem of Sikorav.

We offer several applications: we reprove Thurston's theorem on the existence of a polytope controlling the BNS invariants of a 3-manifold group; we extend this result to free-by-cyclic groups, and the more general descending HNN extensions of free groups. We also show that the BNS invariants of Poincaré duality groups of type F in dimension 3 and groups of deficiency one are determined by a polytope, when the groups are assumed to be agrarian, that is their integral group rings embed in skew-fields. The latter result partially confirms a conjecture of Friedl.

We also deduce the vanishing of the Newton polytopes associated to elements of the Whitehead groups of many groups satisfying the Atiyah conjecture. We use this to show that the $L^2$-torsion polytope of Friedl-Lück is invariant under homotopy. We prove the vanishing of this polytope in the presence of amenability, thus proving a conjecture of Friedl-Lück-Tillmann.

arXiv Invent. Math. (2020) open access talk in Urbana-Champaign and Warwick
12. Appendix to "Amenability of groups is characterized by Myhill's Theorem" by Laurent Bartholdi

We prove a converse to Myhill's "Garden-of-Eden" theorem and obtain in this manner a characterization of amenability in terms of cellular automata: "A group G is amenable if and only if every cellular automaton with carrier G that has gardens of Eden also has mutually erasable patterns."

This answers a question by Schupp, and solves a conjecture by Ceccherini-Silberstein, Machì and Scarabotti.

An appendix by Dawid Kielak proves that group rings without zero divisors are Ore domains precisely when the group is amenable, answering a conjecture attributed to Guba.

arXiv J. Eur. Math. Soc. (2019)
13. On the smallest non-abelian quotient of $\mathrm{Aut}(F_n)$ (with Barbara Baumeister and Emilio Pierro)

We show that the smallest non-abelian quotient of $\mathrm{Aut}(F_n)$ is $\mathrm{PSL}_n(\mathbb Z/2 \mathbb Z)=\mathrm L_n(2)$, thus confirming a conjecture of Mecchia-Zimmermann. In the course of the proof we give an exponential (in $n$) lower bound for the cardinality of a set on which $\mathrm{SAut}(F_n)$, the unique index $2$ subgroup of $\mathrm{Aut}(F_n)$, can act non-trivially. We also offer new results on the representation theory of $\mathrm{SAut}(F_n)$ in small dimensions over small, positive characteristics, and on rigidity of maps from $\mathrm{SAut}(F_n)$ to finite groups of Lie type and algebraic groups in characteristic $2$.

arXiv Proc. Lond. Math. Soc. (2019) open access
14. Appendix (with Damian Sawicki) to "Warped cones, (non-)rigidity, and piecewise properties" by Damian Sawicki

We prove that if a quasi-isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi-isometry of the respective warped cones. For a general quasi-isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi-isometric after taking Cartesian products with suitable powers of the integers.

Secondly, we characterise geometric properties of a group (coarse embeddability into Banach spaces, asymptotic dimension, property A) by properties of the warped cone over an action of this group. These results apply to arbitrary asymptotically faithful coverings, in particular to box spaces. As an application, we calculate the asymptotic dimension of a warped cone, improve bounds by Szabó, Wu, and Zacharias and by Bartels on the amenability dimension of actions of virtually nilpotent groups, and give a partial answer to a question of Willett about dynamic asymptotic dimension.

In the appendix, we justify optimality of the aforementioned result on general quasi-isometries by showing that quasi-isometric warped cones need not come from quasi-isometric groups, contrary to the case of box spaces.

arXiv Proc. Lond. Math. Soc. (2019)
15. Nielsen realisation for untwisted automorphisms of right-angled Artin groups (with Sebastian Hensel)

We prove Nielsen realisation for finite subgroups of the groups of untwisted outer automorphisms of RAAGs in the following sense: given any graph $\Gamma$, and any finite group $G \leqslant \mathrm{U}^0(A_\Gamma)$ $\leqslant \mathrm{Out}^0(A_\Gamma)$, we find a non-positively curved cube complex with fundamental group $A_\Gamma$ on which $G$ acts by isometries, realising the action on $A_\Gamma$.

arXiv Proc. Lond. Math. Soc. (2018) poster
16. Alexander and Thurston norms, and the Bieri-Neumann-Strebel invariants for free-by-cyclic groups (with Florian Funke)

We investigate Friedl-Lück's universal $L^2$-torsion for descending HNN extensions of finitely generated free groups, and so in particular for $F_n$-by-$\mathbb Z$ groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for $3$-manifold groups. We prove that this Thurston semi-norm is an upper bound for the Alexander semi-norm defined by McMullen, as well as for the higher Alexander semi-norms defined by Harvey. The same inequalities are known to hold for $3$-manifold groups. We also prove that the Newton polytopes of the universal $L^2$-torsion of a descending HNN extension of $F_2$ locally determine the Bieri-Neumann-Strebel invariant of the group. We give an explicit means of computing the BNS invariant for such groups. As a corollary, we prove that the Bieri-Neumann-Strebel invariant of a descending HNN extension of $F_2$ has finitely many connected components. When the HNN extension is taken over Fn along a polynomially growing automorphism with unipotent image in $\mathrm{GL}(n,\mathbb Z)$, we show that the Newton polytope of the universal $L^2$-torsion and the BNS invariant completely determine one another. We also show that in this case the Alexander norm, its higher incarnations, and the Thurston norm all coincide.

arXiv Geom. Topol. (2018) poster talk in Bonn and Cambridge
17. Nielsen realisation by gluing: Limit groups and free products (with Sebastian Hensel)

We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel-Mosher and on the outer space of a free product of Guirardel-Levitt, as well as a relative version of the Nielsen realisation theorem, which in the case of free groups answers a question of Karen Vogtmann. We also prove Nielsen realisation for limit groups, and as a byproduct obtain a new proof that limit groups are CAT(0). The proofs rely on a new version of Stallings' theorem on groups with at least two ends, in which some control over the behaviour of virtual free factors is gained.

arXiv Michigan Math. J. (2018)
18. Outer actions of $\mathrm{Out}(F_n)$ on small right-angled Artin groups

We determine the precise conditions under which $\mathrm{SOut}(F_n)$, the unique index two subgroup of $\mathrm{Out}(F_n)$, can act non-trivially via outer automorphisms on a RAAG whose defining graph has fewer than $\frac 1 2 \binom n 2$ vertices.

We also show that the outer automorphism group of a RAAG cannot act faithfully via outer automorphisms on a RAAG with a strictly smaller (in number of vertices) defining graph.

Along the way we determine the minimal dimensions of non-trivial linear representations of congruence quotients of the integral special linear groups over algebraically closed fields of characteristic zero, and provide a new lower bound on the cardinality of a set on which $\mathrm{SOut}(F_n)$ can act non-trivially.

arXiv Algebr. Geom. Topol. (2018)
19. The 6-strand braid group is CAT(0) (with Thomas Haettel and Petra Schwer)

We show that braid groups with at most $6$ strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes and the embeddability of their diagonal links into spherical buildings of type $A$. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCammond.

arXiv Geom. Dedicata (2016) open access photo of the authors
20. Groups with infinitely many ends are not fraction groups

We show that any finitely generated group $F$ with infinitely many ends is not a group of fractions of any finitely generated proper subsemigroup $P$, that is $F$ cannot be expressed as a product $PP^{−1}$. In particular this solves a conjecture of Navas in the positive. As a corollary we obtain a new proof of the fact that finitely generated free groups do not admit isolated left-invariant orderings.

arXiv Groups Geom. Dyn. (2015)
21. Low dimensional free and linear representations of $\mathrm{Out}(F_3)$

We study homomorphisms from $\mathrm{Out}(F_3)$ to $\mathrm{Out}(F_5)$, and $\mathrm{GL}(m,K)$ for $m<7$, where $K$ is a field of characteristic other than $2$ or $3$. We conclude that all $K$-linear representations of dimension at most $6$ of $\mathrm{Out}(F_3)$ factor through $\mathrm{GL}(3,\mathbb Z)$, and that all homomorphisms from $\mathrm{Out}(F_3)$ to $\mathrm{Out}(F_5)$ have finite image.

arXiv J. Group Theory (2015)
22. Outer automorphism groups of free groups: linear and free representations

We study the existence of homomorphisms between $\mathrm{Out}(F_n)$ and $\mathrm{Out}(F_m)$ for $n > 5$ and $m < n(n-1)/2$, and conclude that if $m$ is not equal to $n$ then each such homomorphism factors through the finite group of order $2$. In particular this provides an answer to a question of Bogopol'skii and Puga. In the course of the argument linear representations of $\mathrm{Out}(F_n)$ in dimension less than $n(n+1)/2$ over fields of characteristic zero are completely classified. It is shown that each such representation has to factor through the natural projection from $\mathrm{Out}(F_n)$ to $\mathrm{GL}(n,\mathbb Z)$ coming from the action of $\mathrm{Out}(F_n)$ on the abelianisation of $F_n$. We obtain similar results about linear representation theory of $\mathrm{Out}(F_4)$ and $\mathrm{Out}(F_5)$.

arXiv J. Lond. Math. Soc. (2013)

## Book chapters and theses

1. Non-crossing partitions (with Barbara Baumeister, Kai-Uwe Bux, Friedrich Götze, and Henning Krause)

Non-crossing partitions have been a staple in combinatorics for quite some time. More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups. Also, analogues of the non-crossing partition lattice have been introduced. Here, the classical non-crossing partitions are associated to Coxeter and Artin groups of type $\mathsf{A}_n$, which explains the tight connection to the symmetric groups and braid groups. We shall outline those developments.

arXiv EMS Series of Congress Reports
• Habilitation thesis "Free groups, RAAGs, and their automorphisms" defended at the Bielefeld University
thesis
• Doctoral thesis "Free and linear representations of outer automorphism groups of free groups" supervised by Martin R. Bridson at the University of Oxford

For various values of $n$ and $m$ we investigate homomorphisms $\mathrm{Out}(F_n) \to \mathrm{Out}(F_m)$ and $\mathrm{Out}(F_n)\to \mathrm{GL}_m(K)$, i.e. the free and linear representations of $\mathrm{Out}(F_n)$ respectively.

By means of a series of arguments revolving around the representation theory of finite symmetric subgroups of $\mathrm{Out}(F_n)$ we prove that each homomorphism $\mathrm{Out}(F_n) \to \mathrm{GL}_m(K)$ factors through the natural map $\pi_n \colon \mathrm{Out}(F_n)\to \mathrm{GL}(H_1(F_n,\mathbb Z))\cong \mathrm{GL}_n(\mathbb Z)$ whenever $n= 3,m <7$ and $\mathrm{char}(K)\not\in \{2,3\}$, and whenever $n >5,m <\binom {n+1} 2$ and $\mathrm{char}(K) \not\in \{2,3,...,n+ 1\}$.

We also construct a new infinite family of linear representations of $\mathrm{Out}(F_n)$ (where $n >2$), which do not factor through $\pi_n$. When $n$ is odd these have the smallest dimension among all known representations of $\mathrm{Out}(F_n)$ with this property.

Using the above results we establish that the image of every homomorphism $\mathrm{Out}(F_n) \to \mathrm{Out}(F_m)$ is finite whenever $n= 3$ and $n < m <6$, and of cardinality at most $2$ whenever $n >5$ and $n < m < \binom n 2$. We further show that the image is finite when $\binom n 2 \leqslant m < \binom{n+1} 2$.

We also consider the structure of normal finite index subgroups of $\mathrm{Out}(F_n)$. If $N$ is such then we prove that if the derived subgroup of the intersection of $N$ with the Torelli subgroup $\overline{\mathrm{IA}}_n < \mathrm{Out(F_n)}$ contains some term of the lower central series of $\overline{\mathrm{IA}}_n$ then the abelianisation of $N$ is finite.

thesis Oxford Research Archive

## Grants and prizes

• Whitehead prize of the LMS (2022)
• DFG Grant within SPP 2026 "Property (T)" (2020)
• ERC Starting Grant "Fibring" (2019)
• DFG Grant within SPP 2026 "Parabolics and Invariants" (2017)
• Carl Albert Price of St. Peter's College, Oxford (2008)

## Current teaching

We are meeting on Fridays at 11am.

We will aim at proving two theorems: that the Grigorchuk Group is of intermediate growth, and then that the lamplighter group violates (the old version of) the Strong Atiyah Conjecture.

1. Week 1: Growth of groups, amenability.
2. Week 2: Nilpotent groups.
3. Week 3: Milnor–Wolf Theorem, part 1.
4. Week 4: Milnor–Wolf Theorem, part 2
5. Week 5: Ozawa's proof of Gromov Theorem.
6. Week 6: Grigorchuk's group.
7. Week 7: Lamplighter group.
8. Week 8: The Strong Atiyah Conjecture.

## Past teaching

1. Knot Theory, SoSe 2020, Bielefeld
2. The lecture will now take the form of a reading group. We will be going though Cromwell's book.

The reading plan is as follows:

1. Week 1 (20.04-23.04): Chapter 1.
2. Week 2 (24.04-30.04): Sections 2.1-2.7.
3. Week 3 (1.05-7.05): Sections 2.8-3.4. Here is a book on the connections between knots and number theory.
4. Week 4 (8.04-14.05): Sections 3.5-3.8. Focus: Theorem 3.8.2. Exercises: 3.10.4, 3.10.13.
5. Week 5 (15.05-21.05): Sections 4.1-4.6. Focus: Theorem 4.5.2. Meeting: Wednesday 10:00.
6. Week 6 (22.05-28.05): Sections 4.7-4.10. Warning: Lemma 4.10.2.
7. Week 7 (19.05-4.06): Sections 5.1-5.4.
8. Week 8 (5.05-11.06): Sections 5.5-5.8. Meeting: Wednesday 10:00.
9. Week 9 (12.05-18.06): Sections 6.1-6.5.
10. Week 10 (19.06-25.06): Sections 6.6-6.8. Meeting: Thursday 2.07.20 10:00.

There is a good selection of materials helpful in learning about knots on the website of Gandalf Lechner.

3. $3$-Manifolds, WiSe 2019/2020, Bielefeld
4. We will follow the book "3-Manifolds" by John Hempel.

1. Lecture 1. Piecewise linear topology - introduction.
2. Lecture 2. Basic theorems of piecewise linear topology.
3. Lecture 3. General position.
4. Lecture 4. Two-sided submanifolds.
5. Lecture 5. Heegaard splittings.
6. Lecture 6. Heegaard diagrams.
7. Lecture 7. Lens spaces.
8. Lecture 8. Non-orientable genus 1 splittings.
9. Lecture 9. Connected sums.
10. Lecture 10. Poincaré conjecture.
11. Lecture 11. $\mathbb{S}^2$-bundles over $\mathbb{S}^1$.
12. Lecture 12. Existence of prime decompositions.
13. Lecture 13. Existence of prime decompositions concluded.
5. $L^2$-Invariants, SoSe 2019, Bielefeld
6. We will follow the notes of Holger Kammeyer. The main reference is the book `$L^2$-Invariants: Theory and Applications to Geometry and $K$-Theory' by Wolfgang Lück.

1. Lecture 1. Cellular homology, modules. Reading: Hatcher's book.
2. Lecture 2. Von Neumann algebras, Hilbert modules. Reading: Chapter 1 of Kammeyer's notes.
3. Lecture 3. CW-complexes via pushouts. Reading: Hatcher's book.
4. Lecture 4. $L^2$-homology and Betti numbers for $G$-CW complexes. Reading: Chapter 2.3 of Kammeyer's notes.
5. Lecture 5. Properties of $L^2$-Betti numbers. Reading: Chapter 2.3 of Kammeyer's notes.
6. Lecture 6. Amenability of groups. Reading: survey by Alejandra Garrido.
7. Lecture 7. Vanishing of $L^2$-Betti numbers. Reading: Section 6.4.3 of Lück's book.
8. Lecture 8. Residually-finite groups and Lück's approximation theorem. Reading: Alex Wright's note.
9. Lecture 9. Linnel's skew-field and torsion. Reading: Lück's book.
7. Congruence Subgroup Property, SoSe 2017, Bielefeld
8. Gruppentheorie, WiSe 2016/2017, Bielefeld
9. Hyperbolic Groups, WiSe 2014/2015, Bonn
10. Groups, Trees, and Hyperbolicity, WiSe 2013/2014, Bonn

## Contact

kielak@maths.ox.ac.uk
Office N2.11
Mathematical Institute
University of Oxford
Andrew Wiles Building