Saul Schleimer

Winnowing: local algorithms for document fingerprinting

Digital content is for copying: quotation, revision, plagiarism, and file sharing all create copies. Document fingerprinting is concerned with accurately identifying copying, including small partial copies, within large sets of documents.We introduce the class of local document fingerprinting algorithms, which seems to capture an essential property of any finger-printing technique guaranteed to detect copies. We prove a novel lower bound on the performance of any local algorithm. We also develop winnowing, an efficient local fingerprinting algorithm, and show that winnowings performance is within 33% of the lower bound. Finally, we also give experimental results on Web data, and report experience with MOSS, a widely-used plagiarism detection service.

The geometry of the disk complex

We give a distance estimate for the disk complex. We use the distance estimate to prove that the disk complex is Gromov hyperbolic. As another application of our techniques, we find an algorithm which computes the Hempel distance of a Heegaard splitting, up to an error depending only on the genus.

High distance knots

We construct knots in S-3 with Heegaard splittings of arbitrarily high distance, in any genus. As an application, for any positive integers t and b we find a tunnel number t knot in the three-sphere which has no (t, b) -decomposition.

Uniform hyperbolicity of the curve graph via surgery sequences

We prove that the curve graph C (1) (S) is Gromov- hyperbolic with a constant of hyperbolicity independent of the surface S. The proof is based on the proof of hyperbolicity of the free splitting complex by Handel and Mosher, as interpreted by Hilion and Horbez.

COVERS AND THE CURVE COMPLEX

We provide the first nontrivial examples of quasi-isometric embeddings between curve complexes; these are induced by orbifold covers. This leads to new quasi-isometric embeddings between mapping class groups. As a corollary, in the mapping class group normalizers of finite subgroups are undistorted.

Polynomial-time word problems

We find polynomial-time solutions to the word problem for free-by-cyclic groups, the word problem for automorphism groups of free groups, and the membership problem for the handlebody subgroup of the mapping class group. All of these results follow from observing that automorphisms of the free group strongly resemble straight-line programs, which are widely studied in the theory of compressed data structures. In an effort to be self-contained we give a detailed exposition of the necessary results from computer science.

Trees and mapping class groups

Abstract There is a forgetful map from the mapping class group of a punctured surface to that of the surface with one fewer puncture. We prove that finitely generated purely pseudo-Anosov subgroups of the kernel of this map are convex cocompact in the sense of B. Farb and L. Mosher. In particular, we obtain an affirmative answer to their question of local convex cocompactness of K. Whittleseys group. In the course of the proof, we obtain a new proof of a theorem of I. Kra. We also relate the action of this kernel on the curve complex to a family of actions on trees. This quickly yields a new proof of a theorem of J. Harer.

Efficient computation in groups via compression

We study the compressed word problem: a variant of the word problem for finitely generated groups where the input word is given by a context-free grammar that generates exactly one string. We show that finite extensions and free products preserve the complexity of the compressed word problem. Also, the compressed word problem for a graph group can be solved in polynomial time. These results allow us to obtain new upper complexity bounds for the word problem for certain automorphism groups and group extensions.

Connectivity of the space of ending laminations

We prove that for any closed surface of genus at least four, and any punctured surface of genus at least two, the space of ending laminations is connected. A theorem of E. Klarreich [28, Theorem 1.3] implies that this space is homeomorphic to the Gromov boundary of the complex of curves. It follows that the boundary of the complex of curves is connected in these cases, answering the conjecture of P. Storm. Other applications include the rigidity of the complex of curves and connectivity of spaces of degenerate Kleinian groups.

Distances of Heegaard splittings

J. Hempel [Topology, 2001] showed that the set of distances of the Heegaard splittings (S;V;h n (V)) is unbounded, as long as the stable and unstable laminations of h avoid the closure of V ‰PML(S). Here h is a pseudo-Anosov homeomorphism of a surface S while V is the set of isotopy classes of simple closed curves in S bounding essential disks in a flxed handlebody. With the same hypothesis we show that the distance of the splitting (S;V;h n (V)) grows linearly with n, answering a question of A. Casson. In addition we prove the converse of Hempel’s theorem. Our method is to study the action of h on the curve complex associated to S . We rely heavily on the result, due to H. Masur and Y. Minsky [Invent. Math., 1999], that the curve complex is Gromov hyperbolic.