Past Topology Seminar

For any word w in a free group of rank r>0, and any compact group G, w induces a `word map' from G^r to G by substitutions of elements of G for the letters of w. We may also choose the r elements of G independently with respect to Haar measure on G, and then apply the word map. This gives a random element of G whose distribution depends on w. An interesting observation is that this distribution doesn't change if we change w by an automorphism of the free group. It is a wide open question whether the measures induced by w on compact groups determine w up to automorphisms.
My talk will be mostly about the case G = U(n), the n by n complex unitary matrices. The technical tool we use is a precise formula for the moments of the distribution induced by w on U(n). In the formula, there is a surprising appearance of concepts from infinite group theory, more specifically, Euler characteristics of mapping class groups of surfaces. I'll explain how our formula allows us to make progress on the question described above.
This is joint work with Doron Puder (Tel Aviv).
 

A group is said to satisfy the Tits Alternative if its finitely generated subgroups exhibit a striking dichotomy: they are either "big" (they contain a non-abelian free subgroup) or "small" (they are virtually soluble). Many groups of geometric interest have been shown to satisfy the Tits Alternative: linear groups, mapping class groups of hyperbolic surfaces, etc. In this talk, I will explain how one can use ideas from group actions in negative curvature to prove such a dichotomy. In particular, I will show how one can prove a strengthening of the Tits Alternative for a large class of Artin groups. This is joint work with Piotr Przytycki.

A two-dimensional, minimally Supersymmetric Quantum Field Theory is "nullhomotopic" if it can be deformed to one with spontaneous supersymmetry breaking, including along deformations that are allowed to "flow up" along RG flow lines. SQFTs modulo nullhomotopic SQFTs form a graded abelian group $SQFT_\bullet$. There are many SQFTs with nonzero index; these are definitely not nullhomotopic, and indeed represent nontorision classes in $SQFT_\bullet$. But relations to topological modular forms suggests that $SQFT_\bullet$ also has rich torsion. Based on an analysis of mock modularity and holomorphic anomalies, I will describe explicitly a "secondary invariant" of SQFTs and use it to show that a certain element of $SQFT_3$ has exact order $24$. This work is joint with D. Gaiotto and E. Witten.

We prove that every rational homology cobordism class in the subgroup generated
by lens spaces contains a unique connected sum of lens spaces whose first homology embeds in
any other element in the same class. As a consequence we show that several natural maps to
the rational homology cobordism group have infinite rank cokernels, and obtain a divisibility
condition between the determinants of certain 2-bridge knots and other knots in the same
concordance class. This is joint work with Daniele Celoria and JungHwan Park.

In a recent paper, Basterra, Bobkova, Ponto, Tillmann and Yeakel defined
topological operads with homological stability (OHS) and proved that the
group completion of an algebra over an OHS is weakly equivalent to an
infinite loop space.

In this talk, I shall outline a construction which to an algebra A over
an OHS associates a new infinite loop space. Under mild conditions on
the operad, this space is equivalent as an infinite loop space to the
group completion of A. This generalises a result of Wahl on the
equivalence of the two infinite loop space structures constructed by
Tillmann on the classifying space of the stable mapping class group. I
shall also talk about an application of this construction to stable
moduli spaces of high-dimensional manifolds in thesense of Galatius and
Randal-Williams.

We will discuss a variant of Taubes’s Seiberg-Witten to Gromov theorem in the context of a 4-manifold with cylindrical ends, equipped with a nontrivial harmonic 2-form. This harmonic 2-form is allowed to be asymptotic to 0 on some (but not all) of its ends, and may have nondegenerate zeros along 1-submanifolds. Corollaries include various positivity results; some simple special cases of these constitute a key ingredient in Kutluhan-Lee-Taubes’s proof of HM = HF (Monopole Floer homology equals Heegaard Floer homology). The aforementioned general theorem is motivated by (potential) extensions of the HM = HF and Lee-Taubes’s HM = PFH (Periodic Floer homology) theorems.

X.S. Lin defined an invariant of knots in S^3 by counting represenations 
of the knot group into SU(2) with fixed meridinal holonomy. Lin's 
invariant was subsequently shown to coincide with the Levine-Tristam 
signature. I'll define an analogous total Lin invariant which counts 
repesentations into both SU(2) and SL_2(R). Unlike the SU(2) version, this 
invariant does not (as far as I know) coincide with other known 
invariants. I'll describe some applications to left-orderability of Dehn 
fillings and branched covers, as well as a curious connection with the 
Alexander polynomial. This is joint work with Nathan Dunfield.

I will explain how Lurie‘s approach to L-theory via Poincaré categories can be extended to yield cobordism categories of Poincaré objects à la Ranicki. These categories can be delooped by an iterated Q-construction and the resulting spectrum is a derived version of Grothendieck-Witt-theory.  Its homotopy type can be described in terms of K- and L-theory as conjectured by Hesselholt-Madsen. Furthermore, it has a clean universal property analogous to that of K-theory, localisation sequences in much greater generality than classical Grothendieck-Witt theory, gives a cycle description of Weiss-Williams‘ LA-theory and allows for maps from the geometric cobordism category, refining and unifying various known invariants.

All original material is joint work with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus and W.Steimle.

The large-scale features of groups and spaces are recorded by asymptotic invariants. Examples of asymptotic invariants are the asymptotic cone and, for hyperbolic groups, the Gromov boundary.
In his study of asymptotic cones of connected Lie groups, Yves Cornulier introduced a class of maps called sublinearly biLipschitz equivalences. Like the more traditionnal quasiisometries, sublinearly biLipschitz equivalences are biLipschitz on the large-scale, but unlike quasiisometries, they are generally not coarse. Sublinearly biLipschitz equivalences still induce biLipschitz homeomorphisms between asymptotic cones. In this talk, I will focus on Gromov-hyperbolic groups and show how the Gromov boundary can be used to produce invariants distinguishing them up to sublinearly biLipschitz equivalences when the asymptotic cones do not. I will especially give applications to the large-scale sublinear geometry of hyperbolic Lie groups.
 

The concept of an acylindrically hyperbolic group, introduced by D. Osin, generalizes hyperbolic and relatively hyperbolic groups, and includes many other groups of interest: Out(F_n), n>1, most mapping class groups, directly indecomposable non-cyclic right angled Artin groups, most graph products, groups of deficiency at least 2, etc. Roughly speaking, a group G is acylindrically hyperbolic if there is a (possibly infinite) generating set X of G such that the Cayley graph \Gamma(G,X) is hyperbolic and the action of G on it is "sufficiently nice". Many global properties of hyperbolic/relatively hyperbolic groups have been also proved for acylindrically hyperbolic groups. 
In the talk I will discuss a method which allows to construct a common acylindrically hyperbolic quotient for any countable family of countable acylindrically hyperbolic groups. This allows us to produce acylindrically hyperbolic groups with many unexpected properties.(The talk will be based on joint work with Denis Osin.)
 

 In the late seventies, Casson and Gordon developed several knot invariants that obstruct a knot from being slice, i.e. from bounding a disc in the 4-ball. In this talk, we use twisted Blanchfield pairings to define twisted generalisations of the Levine-Tristram signature function, and describe their relation to the Casson-Gordon invariants. If time permits, we will present some obstructions to algebraic knots being slice. This is joint work with Maciej Borodzik and Wojciech Politarczyk.

I will discuss Agol's proof of the Virtually Fibred Conjecture of
Thurston, focusing on the role played by the `RFRS' property. I will
then show how one can modify parts of Agol's proof by replacing some
topological considerations with a group theoretic statement about
virtual fibring of RFRS groups.
 

In a joint work with Matt Tointon, we study the fine structure of approximate groups. We deduce various applications on growth, isoperimetry and quantitative estimates for the the simple random walk on finite vertex transitive graphs.

A conjecture from the '80s claims that the isometry type of a closed, negatively curved Riemannian manifold should be uniquely determined by the lengths of its closed geodesics. By work of Otal, this is essentially equivalent to the problem of extending cross-ratio preserving maps between Gromov boundaries of simply connected, negatively curved manifolds. Progress on the conjecture has been remarkably slow, with only the 2-dimensional and locally symmetric cases having been solved so far (Otal '90 and Hamenstädt '99).
Still, it is natural to try leaving the world of manifolds and address the conjecture in the general context of non-positively curved metric spaces. We restrict to the class of CAT(0) cube complexes, as their geometry is both rich and well-understood. We introduce a new notion of cross ratio on their horoboundary and use it to provide a full answer to the conjecture in this setting. More precisely, we show that essential, hyperplane-essential cubulations of Gromov-hyperbolic groups are completely determined by their combinatorial length functions. One can also consider non-proper non-cocompact actions of non-hyperbolic groups, as long as the cube complexes are irreducible and have no free faces.
Joint work with J. Beyrer and M. Incerti-Medici.

Stable commutator length (scl) is a well established invariant of elements g in the commutator subgroup (write scl(g)) and has both geometric and algebraic meaning.  A group has a \emph{gap} in stable commutator length if for every non-trivial element g, scl(g) > C for some C > 0.
SCL may be interpreted as an 'algebraic translation length' and such a gap may be thus interpreted an 'algebraic injectivity radius'.
Many classes of groups have such a gap, like hyperbolic groups, mapping class groups, Baumslag-Solitar groups and graph of groups.
In this talk I will show that Right-Angled Artin Groups have the optimal scl-gap of 1/2. This yields a new invariant for the vast class of subgroups of Right-Angled Artin Groups.

Symmetric spaces and lattices are important objects to model spaces in geometry and topology. They have been studied from many different viewpoints. We will concentrate on their coarse geometry view point in this talk. I will first quickly go over some well-known results about quasi-isometry of those spaces. Then I will move to the study about quasi-isometric embeddings. While results in this direction are far less complete and well-studied, there are some rigidity phenomenons still happening here.

In his famous book on partial differential relations Gromov formulates an exercise concerning local deformations of solutions to open partial differential relations. We will explain the content of this fundamental assertion and sketch a proof. 

In the sequel we will apply this to extend local deformations of closed $G_2$ structures, and to construct 
$C^{1,1}$-Riemannian metrics which are positively curved "almost everywhere" on arbitrary manifolds. 

This is joint work with Christian Bär (Potsdam).

We recall the impact of stabilizing a 4-manifold with $S^2 \times S^2$. The corresponding local situation concerns knots in the 3-sphere which bound (nullhomotopic) discs in a stabilized 4-ball. We explain how these discs arise, and discuss bounds on the minimal number of stabilizations needed. Then we compare this minimal number to the 4-genus.
This is joint work with A. Conway.

I will describe an extremely easy construction with formal group laws, and a 
slightly more subtle argument to show that it can be done in a coordinate-free
way with formal groups.  I will then describe connections with a range of other
phenomena in stable homotopy theory, although I still have many more 
questions than answers about these.  In particular, this should illuminate the
relationship between the Lambda algebra and the Dyer-Lashof algebra at the
prime 2, and possibly suggest better ways to think about related things at 
odd primes.  The Morava K-theory of symmetric groups is well-understood
if we quotient out by transfers, but somewhat mysterious if we do not pass
to that quotient; there are some suggestions that dilation will again be a key
ingredient in resolving this.  The ring $MU_*(\Omega^2S^3)$ is another
object for which we have quite a lot of information but it seems likely that 
important ideas are missing; dilation may also be relevant here.
 

The lower central series of a group $G$ is defined by $\gamma_1=G$ and $\gamma_n = [G,\gamma_{n-1}]$. The "dimension series", introduced by Magnus, is defined using the group algebra over the integers: $\delta_n = \{g: g-1\text{ belongs to the $n$-th power of the augmentation ideal}\}$.

It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has $\delta_n\ge\gamma_n$, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with $\delta_4/\gamma_4$ cyclic of order 2. On the positive side, Sjogren showed that $\delta_n/\gamma_n$ is always a torsion group, of exponent bounded by a function of $n$. Furthermore, it was believed (and falsely proven by Gupta) that only $2$-torsion may occur.
In joint work with Roman Mikhailov, we prove however that for every prime $p$ there is a group with $p$-torsion in some quotient $\delta_n/\gamma_n$.
Even more interestingly, I will show that the dimension quotient $\delta_n/gamma_n$ is related to the difference between homotopy and homology: our construction is fundamentally based on the order-$p$ element in the homotopy group $\pi_{2p}(S^2)$ due to Serre.
 

 I will introduce two obstructions for a rational homology 3-sphere to smoothly bound a rational homology 4-ball- one coming from Donaldson's theorem on intersection forms of definite 4-manifolds, and the other coming from correction terms in Heegaard Floer homology. If L is a nonunimodular definite lattice, then using a theorem of Elkies we will show that whether L embeds in the standard definite lattice of the same rank is completely determined by a collection of lattice correction terms, one for each metabolizing subgroup of the discriminant group. As a topological application this gives a rephrasing of the obstruction coming from Donaldson's theorem. Furthermore, from this perspective it is easy to see that if the obstruction to bounding a rational homology ball coming from Heegaard Floer correction terms vanishes, then (under some mild hypotheses) the obstruction from Donaldson's theorem vanishes too.

In this talk I will articulate and contextualize the following sequence of results.

The Bruhat decomposition of the general linear group defines a stratification of the orthogonal group.
Matrix multiplication defines an algebra structure on its exit-path category in a certain Morita category of categories.  
In this Morita category, this algebra acts on the category of n-categories -- this action is given by adjoining adjoints to n-categories. 

This result is extracted from a larger program -- entirely joint with John Francis, some parts joint with Nick Rozenblyum -- which proves the cobordism hypothesis.  

In this talk I will discuss recent joint work with Dominik Gruber where 
we find a reasonable model for random (infinite) Burnside groups, 
building on earlier tools developed by Coulon and Coulon-Gruber.

The free Burnside group with rank r and exponent n is defined to be the 
quotient of a free group of rank r by the normal subgroup generated by 
all elements of the form g^n; quotients of such groups are called 
Burnside groups.  In 1902, Burnside asked whether any such groups could 
be infinite, but it wasn't until the 1960s that Novikov and Adian showed 
that indeed this was the case for all large enough odd n, with later 
important developments by Ol'shanski, Ivanov, Lysenok and others.

In a different direction, when Gromov developed the theory of hyperbolic 
groups in the 1980s and 90s, he observed that random quotients of free 
groups have interesting properties: depending on exactly how one chooses 
the number and length of relations one can typically gets hyperbolic 
groups, and these groups are infinite as long as not too many relations 
are chosen, and exhibit other interesting behaviour.  But one could 
equally well consider what happens if one takes random quotients of 
other free objects, such as free Burnside groups, and that is what we 
will discuss.
 

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 a generalization of this result, allowing for other notions of length. For example, the same asymptotics hold if we put any (singular) Riemannian metric on the surface. The main ingredient in this generalization is to study measures on the space of geodesic currents.