Koji Fujiwara

Bounded cohomology of subgroups of mapping class groups

We show that every subgroup of the mapping class group MCG(S )o f ac ompact surface S is either virtually abelian or it has innite dimensional second bounded cohomology. As an application, we give another proof of the Farb{ Kaimanovich{Masur rigidity theorem that states that MCG(S )d oes not contain a higher rank lattice as a subgroup.

Eigenvalues of Laplacians on a closed Riemannian manifold and its nets

We show that the eigenvalues of the Laplacian of a closed manifold M is approximated in a certain sense by the eigenvalues of the Laplacian of the graph of a I-net in M as n -*oo . Our approximation needs no assumption on M except for dimension.

The Second Bounded Cohomology of a Group Acting on a Gromov-Hyperbolic Space

Suppose a group G acts on a Gromov-hyperbolic space X properly discontinuously. If the limit set L(G) of the action has at least three points, then the second bounded cohomology group of G, H 2 b (G;R) is in nite dimensional. For example, ifM is a complete, pinched negatively curved Riemannian manifold with nite volume, thenH 2 b ( 1 (M);R) is in nite dimensional. As an application, we show that if G is a knot group with G 6 Z, then H 2 b (G;R) is in nite dimensional.

Stable commutator length in word-hyperbolic groups

In this paper we obtain uniform positive lower bounds on the nstable commutator length of elements in word-hyperbolic groups and certain ngroups acting on hyperbolic spaces (namely the mapping class group acting non the complex of curves, and an amalgamated free product acting on an associated nBass-Serre tree). If G is a word-hyperbolic group that is δ-hyperbolic nwith respect to a symmetric generating set S, then there is a positive constant nC depending only on δ and on |S| such that every element of G either nhas a power which is conjugate to its inverse, or else the stable commutator nlength of the element is at least equal to C. By Bavard’s theorem, these lower nbounds on stable commutator length imply the existence of quasimorphisms nwith uniform control on the defects; however, we show how to construct such nquasimorphisms directly. nWe also prove various separation theorems on families of elements in such ngroups, constructing homogeneous quasimorphisms (again with uniform estimates) nwhich are positive on some prescribed element while vanishing on nsome family of independent elements whose translation lengths are uniformly nbounded. nFinally, we prove that the first accumulation point for stable commutator nlength in a torsion-free word-hyperbolic group is contained between 1/12 and n1/2. This gives a universal sense of what it means for a conjugacy class in na hyperbolic group to have a small stable commutator length, and can be nthought of as a kind of “homological Margulis lemma”.

Fixed point sets of parabolic isometries of CAT(0)-spaces

We study the fixed point set in the ideal boundary of a parabolic isometry of a proper CAT(0)-space. We show that the radius of the fixed point set is at most

The second bounded cohomology of an amalgamated free product of groups

pi/2

Maximal and pointwise ergodic theorems for word-hyperbolic groups

, and study its centers. As a consequence, we prove that the set of fixed points is contractible with respect to the Tits topology.

Combable functions, quasimorphisms, and the central limit theorem

We study the second bounded cohomology of an amalgamated free product of groups, and an HNN extension of a group. As an application, we show that a group with infinitely many ends has infinite dimensional second bounded cohomology.

Bounded cohomology with coefficients in uniformly convex Banach spaces

Let denote a word-hyperbolic group, and let S = S 1 denote a nite symmetric set of generators. Let S n = fw : jwj = ng denote the sphere of radius n, where j j denotes the word length on induced by S. De ne n d = 1 #S n P w2S n w, and n = 1 n+1 P n k=0 k . Let (X;B;m) be a probability space on which acts ergodically by measure preserving transformations. We prove a strong maximal inequality in L 2 for the maximal operator f = sup n 0 j n f(x)j. The maximal inequality is applied to prove a pointwise ergodic theorem in L 2 for exponentially mixing actions of , of the following form : n f (x) ! R X fdm almost everywhere and in the L 2 -norm, for every f 2 L 2 (X). As a corollary, for a uniform lattice G, where G is a simple Lie group of real rank one, we obtain a pointwise ergodic theorem for the action of on an arbitrary ergodic G-space. In particular, this result holds when X = G= is a compact homogeneous space, and yields an equidistribution result for sets of lattice points of the form g, for almost every g 2 G. x1 Definitions and statements of results 1.1 De nition of ergodic sequences. Let be a countable group, and let ` 1 ( ) = f = P 2 ( ) : P 2 j ( )j < 1g denote the group algebra. Given any unitary representation of in a Hilbert space H, extend to the group algebra by: ( ) = P 2 ( ) ( ). Denote by H 1 the space of vectors invariant under every ( ), 2 , and by E 1 the orthogonal projection on H 1 . De nition 1.1. Given a unitary representation ( ;H) of , a sequence n 2 ` 1 ( ) is a mean ergodic sequence in H if k ( n )f E 1 fk ! n!1 0 for all f 2 H. 1991 Mathematics Subject Classi cation. 22D40, 28D15, 43A20, 43A62.

The asymptotic dimension of a curve graph is finite

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 nfunction is bicombable if it is Lipschitz in both the left- and right-invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include: n(1) homomorphisms to Z; n(2) word length with respect to a finite generating set; n(3) most known explicit constructions of quasimorphisms (e.g. the Epstein–Fujiwara counting quasimorphisms). nWe show that bicombable functions on word-hyperbolic groups satisfy a central limit theorem: if φ(overbar)_n is the value of φ on a random element of word length n (in a certain sense), there are E and σ for which there is convergence in the sense of distribution n^(-1/2)(φ(overbar)_n - nE)→ 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 λ_(1,2) depending on S_1 and S_2 such that almost every word of length n in the S1 metric has word length n∙λ_(1,2) in the S_2 metric, with error of size O (√n).