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Danny Calegari

Publications and source records attributed to Danny Calegari.

At least 55 records · Page 3Linked to original sources
arXiv2011

Stable W-length

We study stable W-length in groups, especially for W equal to the n-fold commutator gamma_n:=[x_1,[x_2, . . . [x_{n-1},x_n]] . . . ]. We prove that in any perfect group, for any n at least 2 and any element g, the stable commutator length of g is at least as big as 2^{2-n} times the stable gamma_n-length of g. We also establish analogues of Bavard duality for words gamma_n and for beta_2:=[[x,y],[z,w]]. Our proofs make use of geometric properties of the asymptotic cones of verbal subgroups with respect to bi-invariant metrics. In particular, we show that for suitable W, these asymptotic cones contain certain subgroups that are normed vector spaces.

math.GR↗
arXiv2011

Immersed surfaces in the modular orbifold

A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the following stability theorem: for every hyperbolic element of the modular group, the product of this element with a sufficiently large power of a parabolic element is represented by a geodesic that virtually bounds an immersed surface.

math.GT↗
arXiv2010

Scl, sails and surgery

We establish a close connection between stable commutator length in free groups and the geometry of sails (roughly, the boundary of the convex hull of the set of integer lattice points) in integral polyhedral cones. This connection allows us to show that the scl norm is piecewise rational linear in free products of Abelian groups, and that it can be computed via integer programming. Furthermore, we show that the scl spectrum of nonabelian free groups contains elements congruent to every rational number modulo $\mathbb{Z}$, and contains well-ordered sequences of values with ordinal type $ω^ω$. Finally, we study families of elements $w(p)$ in free groups obtained by surgery on a fixed element $w$ in a free product of Abelian groups of higher rank, and show that $\scl(w(p)) \to \scl(w)$ as $p \to \infty$.

math.GR↗
arXiv2010

Chimneys, leopard spots, and the identities of Basmajian and Bridgeman

We give a simple geometric argument to derive in a common manner orthospectrum identities of Basmajian and Bridgeman. Our method also considerably simplifies the determination of the summands in these identities. For example, for every odd integer n, there is a rational function q_n of degree 2(n-2) so that if M is a compact hyperbolic manifold of dimension n with totally geodesic boundary S, there is an identity χ(S) = \sum_i q_n(e^{l_i}) where the sum is taken over the orthospectrum of M. When n=3, this has the explicit form \sum_i 1/(e^{2l_i}-1) = -χ(S)/4.

math.GT↗
arXiv2010

On fibered commensurability

This paper initiates a systematic study of the relation of commensurability of surface automorphisms, or equivalently, fibered commensurability of 3-manifolds fibering over the circle. We show that every hyperbolic fibered commensurability class contains a unique minimal element, whereas the class of Seifert manifolds fibering over the circle consists of a single commensurability class with infinitely many minimal elements. The situation for non-geometric manifolds is more complicated, and we illustrate a range of phenomena that can occur in this context.

math.GT↗
arXiv2010

Bounded cochains on 3-manifolds

We study the large-scale geometry of 3-manifolds with nontrivial 2-dimensional bounded cohomology, with a view to proving a weak version of the geometrization conjecture for such manifolds.

math.GT↗
arXiv2009

Stable commutator length in word-hyperbolic groups

In this paper we obtain uniform positive lower bounds on stable commutator length in word-hyperbolic groups and certain groups acting on hyperbolic spaces (namely the mapping class group acting on the complex of curves, and an amalgamated free product acting on the Bass-Serre tree). If G is a word hyperbolic group which is delta hyperbolic with respect to a symmetric generating set S, then there is a positive constant C depending only on delta and on |S| such that every element of G either has a power which is conjugate to its inverse, or else the stable commutator length is at least equal to C. By Bavard's theorem, these lower bounds on stable commutator length imply the existence of quasimorphisms with uniform control on the defects; however, we show how to construct such quasimorphisms directly. We also prove various separation theorems, constructing homogeneous quasimorphisms (again with uniform estimates) which are positive on some prescribed element while vanishing on some family of independent elements whose translation lengths are uniformly bounded. Finally, we prove that the first accumulation point for stable commutator length in a torsion-free word hyperbolic group is contained between 1/12 and 1/2. This gives a universal sense of what it means for a conjugacy class in a hyperbolic group to have a small stable commutator length, and can be thought of as a kind of "homological Margulis lemma".

math.GR↗
arXiv2009

Combable functions, quasimorphisms, and the central limit theorem

A function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left and right invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include (i) homomorphisms to Z (ii) word length with respect to a finite generating set (iii) most known explicit constructions of quasimorphisms (e.g. the Epstein-Fujiwara counting quasimorphisms) We show that bicombable functions on word-hyperbolic groups satisfy a central limit theorem: if \barϕ_n is the value of ϕon a random element of word length n (in a certain sense), there are E and σfor which there is convergence in the sense of distribution n^{-1/2}(\barϕ_n - nE) \to N(0,σ), where N(0,σ) denotes the normal distribution with standard deviation σ. As a corollary, we show that if S_1 and S_2 are any two finite generating sets for G, there is an algebraic number lambda_{1,2} depending on S_1 and S_2 such that almost every word of length n in the S_1 metric has word length nλ_{1,2} in the S_2 metric, with error of size O(\sqrt{n}).

math.GR↗
arXiv2009

Positivity of the universal pairing in 3 dimensions

Associated to a closed, oriented surface S is the complex vector space with basis the set of all compact, oriented 3-manifolds which it bounds. Gluing along S defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented 3-manifolds. The main result in this paper is that this pairing is positive, i.e. that the result of pairing a nonzero vector with itself is nonzero. This has bearing on the question of what kinds of topological information can be extracted in principle from unitary 2+1 dimensional TQFTs. The proof involves the construction of a suitable complexity function c on all closed 3-manifolds, satisfying a gluing axiom which we call the topological Cauchy-Schwarz inequality, namely that c(AB) <= max(c(AA),c(BB)) for all A,B which bound S, with equality if and only if A=B. The complexity function c involves input from many aspects of 3-manifold topology, and in the process of establishing its key properties we obtain a number of results of independent interest. For example, we show that when two finite volume hyperbolic 3-manifolds are glued along an incompressible acylindrical surface, the resulting hyperbolic 3-manifold has minimal volume only when the gluing can be done along a totally geodesic surface; this generalizes a similar theorem for closed hyperbolic 3-manifolds due to Agol-Storm-Thurston.

math.GT↗
arXiv2009

Universal circles for quasigeodesic flows

We show that if M is a hyperbolic 3-manifold which admits a quasigeodesic flow, then pi_1(M) acts faithfully on a universal circle by homeomorphisms, and preserves a pair of invariant laminations of this circle. As a corollary, we show that the Thurston norm can be characterized by quasigeodesic flows, thereby generalizing a theorem of Mosher, and we give the first example of a closed hyperbolic 3-manifold without a quasigeodesic flow, answering a long-standing question of Thurston.

math.GT↗
arXiv2009

Distortion in transformation groups

We exhibit rigid rotations of spheres as distortion elements in groups of diffeomorphisms, thereby answering a question of J Franks and M Handel. We also show that every homeomorphism of a sphere is, in a suitable sense, as distorted as possible in the group Homeo(S^n), thought of as a discrete group. An appendix by Y de Cornulier shows that Homeo(S^n) has the strong boundedness property, recently introduced by G Bergman. This means that every action of the discrete group Homeo(S^n) on a metric space by isometries has bounded orbits.

math.DS↗
arXiv2009

Stable commutator length is rational in free groups

For any group, there is a natural (pseudo-)norm on the vector space B1 of real (group) 1-boundaries, called the stable commutator length norm. This norm is closely related to, and can be thought of as a relative version of, the Gromov (pseudo)-norm on (ordinary) homology. We show that for a free group, the unit ball of this pseudo-norm is a rational polyhedron. It follows that stable commutator length in free groups takes on only rational values. Moreover every element of the commutator subgroup of a free group rationally bounds an injective map of a surface group. The proof of these facts yields an algorithm to compute stable commutator length in free groups. Using this algorithm, we answer a well-known question of Bavard in the negative, constructing explicit examples of elements in free groups whose stable commutator length is not a half-integer.

math.GR↗
arXiv2009

Faces of the scl norm ball

Let F be the fundamental group of S, where S is a compact, connected, oriented surface with negative Euler characteristic and nonempty boundary. (1) The projective class of the chain \partial S in B_1(F) intersects the interior of a codimension one face of the unit ball in the stable commutator length pseudo-norm. (2) The unique homogeneous quasimorphism on F dual to this face (up to scale and elements of H^1) is the rotation quasimorphism associated to the action of F on the ideal boundary of the hyperbolic plane, coming from a hyperbolic structure on S. These facts follow from the fact that every homologically trivial 1-chain in S rationally cobounds an immersed surface with a sufficiently large multiple of the boundary. This is true even if S has no boundary.

math.GT↗
arXiv2008

Large scale geometry of commutator subgroups

Let G be a finitely presented group, and G' its commutator subgroup. Let C be the Cayley graph of G' with all commutators in G as generators. Then C is large scale simply connected. Furthermore, if G is a torsion-free nonelementary word-hyperbolic group, C is one-ended. Hence (in this case), the asymptotic dimension of C is at least 2.

math.GR↗
arXiv2008

The Euler class of planar groups

This is an exposition of the homological classification of actions of surface groups on the plane, in every degree of smoothness.

math.DS↗