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Abstract. Let G = P U(1, d) be the group of holomorphic isometries of complex hyperbolic space H d C . The latter is a Kähler manifold with constant negative holomorphic sectional curvature. We call a finitely generated discrete group Γ = g 1 , . . . , gn ⊂ G a marked classical Schottky group of rank n if there is a fundamental polyhedron for G whose sides are equidistant hypersurfaces which are disjoint and not asymptotic, and for which g 1 , . . . , gn are side-pairing transformations. We consider smooth families of such groups Γt = g 1,t , . . . , gn,t with g j,t depending smoothly (C 1 ) on t whose fundamental polyhedra also vary smoothly. The groups Γt are all algebraically isomorphic to the free group in n generators, i.e. there are canonical isomorphisms φt : Γ 0 → Γt. We shall construct a homeomorphism Ψt ofwhich is equivariant with respect to these groups:which is quasiconformal on ∂H d C with respect to the Heisenberg metric, and which is symplectic in the interior. As a corollary, the limit sets of such Schottky groups of equal rank are quasiconformally equivalent to each other.The main tool for the construction is a time-dependent Hamiltonian vector field used to define a diffeomorphism, mapping D 0 onto Dt, where Dt is a fundamental domain of Γt. In two steps, this is extended equivariantly toThe method yields similar results for real hyperbolic space, while the analog for the other rank-one symmetric spaces of noncompact type cannot hold.

Researchers working in technical disciplines wishing to search for information related to a particular mathematical expression cannot effectively do so with a text-based search engine unless they know appropriate text keywords. To overcome this difficulty, we demonstrate a math-aware search engine, which extends the capability of existing text search engines to search mathematical content.Our search engine is composed of a MathFind processing layer implemented on top of a typical text-based search engine layer. Our prototype piggybacks upon the Apache Lucene Search API, a modified vector space model-based text retrieval system.The MathFind layer of the search engine analyzes expressions in MathML, an XML standard for representing mathematical notation. The process decomposes the mathematical expression into a sequence of text-encoded math fragments. These math fragments are analogous to words in a text document. Math fragments combined with text content serve as input to the textsearch engine. At query time, a graphical equation editor is used to enter a math query, which is internally represented in MathML. The math-processing layer converts the MathML query into a sequence of text-encoded math query terms, which form the basis of a text query performed by the underlying text-search engine. To overcome the ambiguity in the presentation of an expression, MathML input is normalized before processing. [1,2] The current implementation has the following features • Indexes variety of document formats: text + MathML, XHTML + MathML, DocBook + MathML, and via conversion, LaTeX, MS Word, and Mathematica notebooks.• The search engine retrieves ranked documents based on similarity to both math and text queries.• The system is capable of interpreting wild card queries in math expressions analogous to text wild queries.• Math query terms can be highlighted in the retrieved documents (cached) to help users locate matched expressions.