Stable commutator length is rational in free groups

Calegari, Danny

doi:10.1090/s0894-0347-09-00634-1

Cited by 49 publications

(83 citation statements)

References 15 publications

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“…Conversely, since S has boundary (by assumption) and therefore 1 .S/ is free, any rational chain C admits an extremal surface, by the Rationality Theorem from [6] (or see Theorem 4.18 of [5]). Hence scl D area.C /=4 if and only if C bounds a positive immersed surface.…”

Section: Rotation Quasimorphismmentioning

confidence: 99%

“…Thus scl defines a pseudo-norm on B In [6] an algorithm is described to compute scl on elements of B H 1 .F / where F is a free group. The program scallop (source available at [4]) implements a polynomialtime version of this algorithm, described in [5, Sections 4.1.7-4.1.8].…”

Section: Scl As a Pseudo-normmentioning

confidence: 99%

“…In finite dimensions, L 1 and L 1 norms are piecewise linear functions, and their unit balls are rational convex polyhedra. Broadly speaking, the main discovery of Calegari [6] is that in free groups (and certain groups derived from free groups in simple ways), computing scl reduces to a finite dimensional linear programming problem, and therefore the unit ball of the scl pseudo-norm on B H 1 .F I R/ is a rational convex polyhedron; ie for every finite dimensional rational vector subspace V of B 1 .F I R/, the unit ball of the scl pseudo-norm restricted to V is a finite-sided rational convex polyhedron (compare with the well-known example of the Gromov-Thurston norm on H 2 of a 3-manifold; see Thurston [21]). …”

Section: Introductionmentioning

confidence: 99%

“…The connection with stable commutator length is as follows: from the main theorem of [6] it follows that in an oriented hyperbolic surface S with boundary, a rational 1-chain C bounds an immersed surface if and only if scl.C / D rot S .C /=2, where rot S is as above (this is Proposition 3.8 below). Hence Theorem C implies that every chain C in B H 1 .…”

Section: Introductionmentioning

confidence: 99%

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Faces of the scl norm ball

Calegari

2009

Geom. Topol.

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Let F = π 1 (S) where S is a compact, connected, oriented surface with χ(S) < 0 and nonempty boundary.(1) The projective class of the the chain ∂S ∈ B H 1 (F ) intersects the interior of a codimension one face π S of the unit ball in the stable commutator length norm on B H 1 (F ). ( 2) The unique homogeneous quasimorphism on F dual to π S (up to scale and elements of H 1 (F )) is the rotation quasimorphism associated to the action of π 1 (S) 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 C in S rationally cobounds an immersed surface with a sufficiently large multiple of ∂S. This is true even if S has no boundary.

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Section: Rotation Quasimorphismmentioning

confidence: 99%

Section: Scl As a Pseudo-normmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Faces of the scl norm ball

Calegari

2009

Geom. Topol.

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“…The function rot can be computed by an exponent sum for a conjugacy class expressed as a product of R's and L's, and scl can be computed using the algorithms described in [4] (implemented on the program scallop, available from [7]) and in [6], §4.2.5.…”

Section: Resultsmentioning

confidence: 99%

Immersed surfaces in the modular orbifold

Calegari

Louwsma

2011

Proc. Amer. Math. Soc.

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Abstract. 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.

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Matrix group integrals, surfaces, and mapping class groups I: U(n)

Magee

Puder

2019

Invent. math.

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Let w be a word in the free group on r generators. The expected value of the trace of the word in r independent Haar elements of O(n) gives a function T r O w (n) of n. We show that T r O w (n) has a convergent Laurent expansion at n = ∞ involving maps on surfaces and L 2 -Euler characteristics of mapping class groups associated to these maps. This can be compared to known, by now classical, results for the GUE and GOE ensembles, and is similar to previous results concerning U (n), yet with some surprising twists.A priori to our result, T r O w (n) does not change if w is replaced with α(w) where α is an automorphism of the free group. One main feature of the Laurent expansion we obtain is that its coefficients respect this symmetry under Aut(F r ).As corollaries of our main theorem, we obtain a quantitative estimate on the rate of decay of T r O w (n) as n → ∞, we generalize a formula of Frobenius and Schur, and we obtain a universality result on random orthogonal matrices sampled according to words in free groups, generalizing a theorem of Diaconis and Shahshahani.Our results are obtained more generally for a tuple of words w 1 , . . . , w , leading to functions T r O w1,...,w . We also obtain all the analogous results for the compact symplectic groups Sp(n) through a rather mysterious duality formula.

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Stable commutator length is rational in free groups

Cited by 49 publications

References 15 publications

Faces of the scl norm ball

Faces of the scl norm ball

Immersed surfaces in the modular orbifold

Matrix group integrals, surfaces, and mapping class groups I: U(n)

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