Abstract
We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is a unique path that is tree reduced. The set of these paths is the Reduced Path Group. It is a continuous analogue of the group of reduced words. The signature of the path is a power series whose coefficients are certain tensor valued definite iterated integrals of the path. We identify the paths with trivial signature as the tree-like paths, and prove that two paths are in tree-like equivalence if and only if they have the same signature. In this way, we extend Chen’s theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for parametrisation in the general setting. It is suggestive to think of this result as a noncommutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. As a second theme we give quantitative versions of Chen’s theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra.
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@incollection {cfkp, MRKEY = {1491098},
AUTHOR = {Cannon, James W. and Floyd, William J. and Kenyon, Richard and Parry, Walter R.},
TITLE = {Hyperbolic Geometry},
BOOKTITLE = {Flavors of Geometry},
SERIES = {Math. Sci. Res. Inst. Publ.},
NUMBER = {31},
PAGES = {59--115},
PUBLISHER = {Cambridge Univ. Press},
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YEAR = {1997},
MRCLASS = {57M50 (57N10)},
MRNUMBER = {99c:57036},
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} -
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MRCLASS = {22.00},
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} -
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author={Fawcett, T.A.},
TITLE={Ph.D. thesis},
NOTE={Mathematical Institute, University of Oxford, 2002},
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BOOKTITLE={Oxford Math. Monogr.},
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@book {LyStFlour, MRKEY = {2314753},
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PAGES = {xviii+109},
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}