Let H < PSL 2 ( Z ) be a finite index normal subgroup which is contained in a principal congruence subgroup, and let Φ ( H ) ≠ H denote a term of the lower central series or the derived series of H . In this paper, we prove that the commensurator of Φ ( H ) in PSL 2 ( R ) is discrete. We thus obtain a natural family of thin subgroups of PSL 2 ( R ) whose commensurators are discrete, establishing some

In an earlier paper we introduced rectangular diagrams of surfaces and showed that any isotopy class of a surface in the three-sphere can be presented by a rectangular diagram. Here we study transformations of those diagrams and introduce basic moves that allow the transition between diagrams representing isotopic surfaces. We also introduce more general combinatorial objects called mirror diagrams

We show that a spectral sequence developed by Lipshitz and Treumann, for application to Heegaard Floer theory, converges to a localized form of topological Hochschild homology with coefficients. This allows us to show that the target of this spectral sequence can be identified with Hochschild homology when the topological Hochschild homology is torsion-free as a module over THH ∗ ( F 2 ) , parallel

We prove that every symplectic 4‐manifold admits a trisection that is compatible with the symplectic structure in the sense that the symplectic form induces a Weinstein structure on each of the three sectors of the trisection. Along the way, we show that a (potentially singular) symplectic braided surface in CP 2 can be symplectically isotoped into bridge position. This paper relies extensively on

We prove a concordance version of the 4‐dimensional light bulb theorem for π 1 ‐negligible compact orientable surfaces, where there is a framed but not necessarily embedded dual sphere. That is, we show that if F 0 and F 1 are such surfaces in a 4‐manifold X that are homotopic and there exists an immersed framed 2‐sphere G in X intersecting F 0 geometrically once, then F 0 and F 1 are concordant if

We find conditions under which a non‐orientable closed surface smoothly embedded into an orientable 4 ‐manifold X can be represented by a connected sum of an embedded closed surface in X and an unknotted projective plane in a 4 ‐sphere. This allows us to extend the Gabai 4 ‐dimensional light bulb theorem and the Auckly–Kim–Melvin–Ruberman–Schwartz ‘one is enough’ theorem to the case of non‐orientable surfaces

We show that non‐elliptic prime 3‐manifolds satisfy integral approximation for the simplicial volume, that is, that their simplicial volume equals the stable integral simplicial volume. The proof makes use of integral foliated simplicial volume and tools from ergodic theory.

For a punctured surface S , we characterize the representations of its fundamental group into PSL 2 ( C ) that arise as the monodromy of a meromorphic projective structure on S with poles of order at most two and no apparent singularities. This proves the analogue of a theorem of Gallo–Kapovich–Marden concerning C P 1 ‐structures on closed surfaces, and settles a long‐standing question about characterizing

We prove a law of large numbers for the volumes of families of random hyperbolic mapping tori and Heegaard splittings providing a sharp answer to a conjecture of Dunfield and Thurston.

We use basic tools of descriptive set theory to prove that a closed set S of marked groups has 2 ℵ 0 quasi‐isometry classes, provided that every non‐empty open subset of S contains at least two non‐quasi‐isometric groups. It follows that every perfect set of marked groups having a dense subset of finitely presented groups contains 2 ℵ 0 quasi‐isometry classes. These results account for most known constructions

We study geometric properties of stabilisers in the handlebody group. We find that stabilisers of meridians are undistorted, while stabilisers of primitive curves or annuli are exponentially distorted for large enough genus.

The complement of an arrangement of hyperplanes in C n has a natural bordification to a manifold with corners formed by removing (or “blowing up”) tubular neighborhoods of the hyperplanes and certain of their intersections. When the arrangement is the complexification of a real simplicial arrangement, the bordification closely resembles Harvey's bordification of moduli space. We prove that the faces

We generalize the Mahowald invariant to the R ‐motivic and C 2 ‐equivariant settings. For all i > 0 with i ≡ 2 , 3 mod 4 , we show that the R ‐motivic Mahowald invariant of ( 2 + ρ η ) i ∈ π 0 , 0 R ( S 0 , 0 ) contains a lift of a certain element in Adams' classical v 1 ‐periodic families, and for all i > 0 , we show that the R ‐motivic Mahowald invariant of η i ∈ π i , i R ( S 0 , 0 ) contains a

Let M be a smooth projective variety and D an ample normal crossings divisor. From topological data associated to the pair ( M , D ) , we construct, under assumptions on Gromov–Witten invariants, a series of distinguished classes in symplectic cohomology of the complement X = M ∖ D . Under further ‘topological’ assumptions on the pair, these classes can be organized into a log(arithmic) PSS morphism

We compute mod ( p , v 1 ) topological Hochschild homology of the connective cover of the K ( 1 ) ‐local sphere spectrum for all primes p ⩾ 3 . This is accomplished using a May‐type spectral sequence in topological Hochschild homology constructed from a filtration of a commutative ring spectrum.

For a finite group G , we define an equivariant cobordism category C d G . Objects of the category are ( d − 1 ) ‐dimensional closed smooth G ‐manifolds and morphisms are smooth d ‐dimensional equivariant cobordisms. We identify the homotopy type of its classifying space (that is, geometric realization of its simplicial nerve) as the fixed points of the infinite loop space of a certain equivariant

We show that quadratic and symmetric L ‐theory of the integers are related by Anderson duality and that both spectra split integrally into the L ‐theory of the real numbers and a generalised Eilenberg–Mac Lane spectrum. As a consequence, we obtain a corresponding splitting of the space G / Top . Finally, we prove analogous results for the genuine L‐spectra recently devised for the study of Grothendieck–Witt theory

The homotopy category of the bordism category h C d has as objects closed oriented ( d − 1 ) ‐manifolds and as morphisms diffeomorphism classes of d ‐dimensional bordisms. Using a new fiber sequence for bordism categories, we compute the classifying space of h C d for d = 1 , exhibiting it as a circle bundle over Ω ∞ − 2 CP − 1 ∞ .

We consider exact fillings with vanishing first Chern class of asymptotically dynamically convex (ADC) manifolds. We construct two structure maps on the positive symplectic cohomology and prove that they are independent of the filling for ADC manifolds. The invariance of the structure maps implies that the vanishing of symplectic cohomology and the existence of symplectic dilations are properties independent

Suppose G is a 1‐ended finitely generated group that is hyperbolic relative to P, a finite collection of 1‐ended finitely generated proper subgroups of G . Our main theorem states that if the boundary ∂ ( G , P ) has no cut point, then G has semistable fundamental group at ∞ . Under mild conditions on G and the members of P, the 1‐ended hypotheses and the no cut point condition can be eliminated to

Let ( S 2 , ω ) be a symplectic sphere, and let τ : S 2 → S 2 be an anti‐symplectic involution of ( S 2 , ω ) . We consider the product ( S 2 , ω ) × ( S 2 , ω ) endowed with the anti‐symplectic involution τ × τ , and study the space of monotone anti‐invariant symplectic forms on this four‐manifold. We show that this space is disconnected. In addition, during the course of the proof, we produce a diffeomorphism

Kronheimer and Mrowka introduced a new knot invariant, called s ♯ , which is a gauge theoretic analogue of Rasmussen's s invariant. In this article, we compute Kronheimer and Mrowka's invariant for some classes of knots, including algebraic knots and the connected sums of quasi‐positive knots with non‐trivial right‐handed torus knots. These computations reveal some unexpected phenomena: we show that

We prove that the group of homeomorphisms of the circle introduced by the author with Justin Moore is of type F ∞ . This makes the group the first example of a type F ∞ group which is nonamenable and does not contain nonabelian free subgroups. To prove our result, we provide a certain generalisation of cube complexes, which we refer to as cluster complexes. We also obtain a computable normal form,

Victor Snaith gave a construction of periodic complex bordism by inverting the Bott element in the suspension spectrum of B U . This presents an E ∞ structure on periodic complex bordism by different means than the usual Thom spectrum definition of the E ∞ ‐ring M U P . Here, we prove that these two E ∞ ‐rings are in fact different, though the underlying E 2 ‐rings are equivalent. Nonetheless, we prove

Given a connected cobordism between two knots in the 3‐sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge index and the band‐unlinking number of a knot, the fusion number of a ribbon knot

Let X be a nonsingular projective algebraic variety over C , and let M ¯ g , n , β ( X ) be the moduli space of stable maps f : ( C , x 1 , … , x n ) → X from genus g , n ‐pointed curves C to X of degree β . Let S be a line bundle on X . Let A = ( a 1 , ⋯ , a n ) be a vector of integers which satisfy ∑ i = 1 n a i = ∫ β c 1 ( S ) . Consider the following condition: the line bundle f ∗ S has a meromorphic

For an ample line bundle L on a complete toric surface X , we consider the subset V L ⊂ | L | of irreducible, nodal, rational curves contained in the smooth locus of X . We study the monodromy map from the fundamental group of V L to the permutation group on the set of nodes of a reference curve C ∈ V L . We identify a certain obstruction map Ψ X defined on the set of nodes of C and show that the image

For a suitable choice of the cube category, we construct a Grothendieck topology on it such that sheaves with respect to this topology are exactly simplicial sets (thus establishing simplicial sets as a reflective subcategory of cubical sets). We then extend the construction of the homotopy coherent nerve to cubical categories and establish an analogue of Lurie's straightening–unstraightening construction

We introduce a generalization of oriented tangles, which are still called tangles, so that they are in one‐to‐one correspondence with the sutured manifolds. We define cobordisms between sutured manifolds (tangles) by generalizing cobordisms between oriented tangles. For every commutative algebra A over Z / 2 Z , we define A - Tangles to be the category consisting of A ‐tangles, which are balanced tangles

We prove a generalization of the classical connectivity theorem of Blakers–Massey, valid in an arbitrary higher topos and with respect to an arbitrary modality, that is, a factorization system ( L , R ) in which the left class is stable by base change. We explain how to rederive the classical result, as well as the recent generalization of Chachólski, Scherer and Werndli (Ann. Inst. Fourier 66 (2016)

In this paper we complete a chain of explicit Quillen equivalences between the model category for Θ n + 1 ‐spaces and the model category of small categories enriched in Θ n ‐spaces. The Quillen equivalences given here connect Segal category objects in Θ n ‐spaces, complete Segal objects in Θ n ‐spaces, and Θ n + 1 ‐spaces.

We characterize groups admitting Anosov representations into SL ( 3 , R ) , projective Anosov representations into SL ( 4 , R ) , and Borel Anosov representations into SL ( 4 , R ) . More generally, we obtain bounds on the cohomological dimension of groups admitting P k ‐Anosov representations into SL ( d , R ) and offer several characterizations of Benoist representations.

We produce for each tropical hypersurface V ( ϕ ) ⊂ Q = R n a Lagrangian L ( ϕ ) ⊂ ( C ∗ ) n whose moment map projection is a tropical amoeba of V ( ϕ ) . When these Lagrangians are admissible in the Fukaya–Seidel category, we show that they are unobstructed objects of the Fukaya category, and mirror to sheaves supported on complex hypersurfaces in a toric mirror.

We study a generalized cusp C that is diffeomorphic to [ 0 , ∞ ) times a closed Euclidean manifold. Geometrically, C is the quotient of a properly convex domain in R P n by a lattice, Γ , in one of a family of affine Lie groups G ( ψ ) , parameterized by a point ψ in the (dual closed) Weyl chamber for SL ( n + 1 , R ) , and Γ determines the cusp up to equivalence. These affine groups correspond to

Let K be a knot in a rational homology sphere M . This paper investigates the question of when modifying K by adding m > 0 half‐twists to two oppositely oriented strands, while keeping the rest of K fixed, produces a knot isotopic to K . Such a two‐strand twist of order m , as we define it, is a generalized crossing change when m is even and a non‐coherent band surgery when m = ± 1 . A cosmetic two‐strand

Given an n ‐component link L in any 3‐manifold M , the space L ⊂ ( Q ∪ { ∞ } ) n of rational surgery slopes yielding L‐spaces is already fully characterized in joint work by the author when n = 1 and L is nontrivial. For n > 1 , however, there are no previous results for L as a rational subspace, and only limited results for integer surgeries L ∩ Z n on S 3 . Herein, we provide the first nontrivial

We introduce a collection of 1 2 ‐ π 1 ‐null four‐dimensional surgery problems. This is an intermediate notion between the classically studied universal surgery models and the π 1 ‐null kernels which are known to admit a solution in the topological category. Using geometric applications of the group‐theoretic 2‐Engel relation, we show that the 1 2 ‐ π 1 ‐null surgery problems are universal, in the

We define the stabilizing number sn ( K ) of a knot K ⊂ S 3 as the minimal number n of S 2 × S 2 connected summands required for K to bound a null‐homologous locally flat disk in D 4 # n S 2 × S 2 . This quantity is defined when the Arf invariant of K is zero. We show that sn ( K ) is bounded below by signatures and Casson–Gordon invariants and bounded above by the topological 4‐genus g 4 top ( K )

Let Γ g , 1 m be the mapping class group of the orientable surface Σ g , 1 m of genus g with one parametrized boundary curve and m permutable punctures; when m = 0 we omit it from the notation. Let β m ( Σ g , 1 ) be the braid group on m strands of the surface Σ g , 1 . We prove that H ∗ ( Γ g , 1 m ; Z 2 ) ≅ H ∗ ( Γ g , 1 ; H ∗ ( β m ( Σ g , 1 ) ; Z 2 ) ) . The main ingredient is the computation of

Generalising results of Razborov and Safin, and answering a question of Button, we prove that for every hyperbolic group there exists a constant α > 0 such that for every finite subset U that is not contained in a virtually cyclic subgroup | U n | ⩾ ( α | U | ) [ ( n + 1 ) / 2 ] . Similar estimates are established for groups acting acylindrically on trees or hyperbolic spaces.

We show that a finitely generated subgroup of Diff ω ( S 1 ) that is expanding and locally discrete in the analytic category is analytically conjugated to a uniform lattice in PGL ∼ 2 k ( R ) acting on the k th covering of R P 1 for a certain integer k > 0 .

Based on recent advances on the relation between geometry and representation theory, we propose a new approach to elliptic Schubert calculus. We study the equivariant elliptic characteristic classes of Schubert varieties of the generalized full flag variety G / B . For this first we need to twist the notion of elliptic characteristic class of Borisov–Libgober by a line bundle, and thus allow the elliptic

We correct a mistake regarding almost complex structures on Hilbert schemes of points in surfaces in Hendricks, Lipshitz and Sarkar (J. Topol. 9 (2016) 1153–1236). The error does not affect the main results of the paper, and only affects the proofs of invariance of equivariant symplectic Khovanov homology and reduced symplectic Khovanov homology. We give an alternate proof of the invariance of equivariant

We study the relationship on Weinstein domains given by Weinstein cobordism. Our main result is that any finite collection of high‐dimensional Weinstein domains with the same topology is Weinstein subdomains of a ‘maximal’ Weinstein domain also with the same topology. As applications, we construct many new exotic Weinstein structures, for example, exotic cotangent bundles containing many closed regular

We explain some interesting relations in the degree 3 bounded cohomology of surface groups. Specifically, we show that if two faithful Kleinian surface group representations are quasi‐isometric, then their bounded fundamental classes are the same in bounded cohomology. This is novel in the setting that one end is degenerate, while the other end is geometrically finite. We also show that a difference

We provide a smoothening criterion for group actions on manifolds by singular diffeomorphisms. We prove that if a countable group Γ has the fixed point property FW for walls (for example, if it has property ( T ) ), every aperiodic action of Γ by diffeomorphisms that are of class C r with countably many singularities is conjugate to an action by true diffeomorphisms of class C r on a homeomorphic (possibly

Let f : C ̂ → C ̂ be a postcritically finite rational map, and let Q ( C ̂ ) be the space of meromorphic quadratic differentials on C ̂ with simple poles. We study the set of eigenvalues of the pushforward operator f ∗ : Q ( C ̂ ) → Q ( C ̂ ) . In particular, we show that when f : C → C is a unicritical polynomial of degree D with periodic critical point, the eigenvalues of f ∗ : Q ( C ̂ ) → Q ( C

We construct taut foliations in every closed 3‐manifold obtained by r ‐framed Dehn surgery along a positive 3‐braid knot K in S 3 , where r < 2 g ( K ) − 1 and g ( K ) denotes the Seifert genus of K . This confirms a prediction of the L‐space Conjecture. For instance, we produce taut foliations in every non‐L‐space obtained by surgery along the pretzel knot P ( − 2 , 3 , 7 ) , and indeed along every

If Σ and Σ ′ are homotopic embedded surfaces in a 4‐manifold, then they may be related by a regular homotopy (at the expense of introducing double points) or by a sequence of stabilisations and destabilisations (at the expense of adding genus). This naturally gives rise to two integer‐valued notions of distance between the embeddings: the singularity distance d sing ( Σ , Σ ′ ) and the stabilisation

We specify exterior generators in π ∗ T H H ( M U ) = π ∗ ( M U ) ⊗ E ( λ n ′ ∣ n ⩾ 1 ) and π ∗ T H H ( B P ) = π ∗ ( B P ) ⊗ E ( λ n ∣ n ⩾ 1 ) , and calculate the action of the σ ‐operator on these graded rings. In particular, σ ( λ n ′ ) = 0 and σ ( λ n ) = 0 , while the actions on π ∗ ( M U ) and π ∗ ( B P ) are expressed in terms of the right units η R in the Hopf algebroids ( π ∗ ( M U ) , π ∗

Surgery exact triangles in various 3‐manifold Floer homology theories provide an important tool in studying and computing the relevant Floer homology groups. These exact triangles relate the invariants of 3‐manifolds, obtained by three different Dehn surgeries on a fixed knot. In this paper, the behavior of SU ( N ) ‐instanton Floer homology with respect to Dehn surgery is studied. In particular, it

We define a differential graded algebra for Legendrian graphs and tangles in the standard contact Euclidean three space. This invariant is defined combinatorially by using ideas from Legendrian contact homology. The construction is distinguished from other versions of Legendrian contact algebra by the vertices of Legendrian graphs. A set of countably many generators and a generalized notion of equivalence

We define the Bianchi–Massey tensor of a topological space X to be a linear map B → H ∗ ( X ) , where B is a subquotient of H ∗ ( X ) ⊗ 4 determined by the algebra H ∗ ( X ) . We then prove that if M is a closed ( n − 1 ) ‐connected manifold of dimension at most 5 n − 3 (and n ⩾ 2 ) then its rational homotopy type is determined by its cohomology algebra and Bianchi–Massey tensor, and that M is formal

We extend techniques due to Pardon to show that there is a lower bound on the distortion of a knot in R 3 proportional to the minimum of the bridge distance and the bridge number of the knot. We also exhibit an infinite family of knots for which the minimum of the bridge distance and the bridge number is unbounded and Pardon's lower bound is constant.

Let G be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semi‐infinite orbits in the affine Grassmannian Gr G . We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semi‐infinite orbits with U ( n ∨ ) (the universal enveloping algebra of the positive

We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using

Let L be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo‐holomorphic disks with boundaries on L , one can define a Maurer–Cartan element of a Lie bracket operation in string topology (the loop bracket) defined at chain level. This observation is due to Fukaya, who also pointed out its important

We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartment classes. Assuming the generalized Riemann hypothesis, this shows that the Steinberg module of a number ring is generated by integral apartment classes if and only if the ring is Euclidean. We also construct new cohomology classes in the top‐dimensional

We relate the recognition principle for infinite P 1 ‐loop spaces to the theory of motivic fundamental classes of Déglise, Jin and Khan. We first compare two kinds of transfers that are naturally defined on cohomology theories represented by motivic spectra: the framed transfers given by the recognition principle, which arise from Voevodsky's computation of the Nisnevish sheaf associated with A n /

We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3‐Calabi–Yau triangulated category. The motivation/application is that the result will be used to show that the (principal component of) space of stability