Abstract
We introduce the notions of treelike path and treelike 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 treelike paths, and prove that two paths are in treelike 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.

[cfkp] J. W. Cannon, W. J. Floyd, R. Kenyon, and W. R. Parry, "Hyperbolic Geometry," in Flavors of Geometry, Cambridge: Cambridge Univ. Press, 1997, pp. 59115.
@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 = {59115},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1997},
MRCLASS = {57M50 (57N10)},
MRNUMBER = {99c:57036},
MRREVIEWER = {Lee Mosher},
ZBLNUMBER = {0899.51012},
} 
[KTCHENone] K. Chen, "Integration of paths—a faithful representation of paths by noncommutative formal power series," Trans. Amer. Math. Soc., vol. 89, pp. 395407, 1958.
@article {KTCHENone, MRKEY = {0106258},
AUTHOR = {Chen, Kuosai},
TITLE = {Integration of pathsa faithful representation of paths by noncommutative formal power series},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {89},
YEAR = {1958},
PAGES = {395407},
ISSN = {00029947},
MRCLASS = {22.00},
MRNUMBER = {21 \#4992},
MRREVIEWER = {J. C. Moore},
DOI = {10.2307/1993193},
ZBLNUMBER = {0097.25803},
} 
[fawcett] T. A. Fawcett, Ph.D. thesis.
@misc{fawcett,
author={Fawcett, T.A.},
TITLE={Ph.D. thesis},
NOTE={Mathematical Institute, University of Oxford, 2002},
} 
[HLnotes] B. M. Hambly and T. J. Lyons, Some notes on trees and paths.
@misc{HLnotes,
author={Hambly, B.M. and Lyons, T.J.},
TITLE={Some notes on trees and paths},
NOTE={http://arxiv.org/abs/0809.1365},
} 
[snake] J. Le Gall, "Brownian excursions, trees and measurevalued branching processes," Ann. Probab., vol. 19, iss. 4, pp. 13991439, 1991.
@article {snake, MRKEY = {1127710},
AUTHOR = {Le Gall, JeanFran{ç}ois},
TITLE = {Brownian excursions, trees and measurevalued branching processes},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {19},
YEAR = {1991},
NUMBER = {4},
PAGES = {13991439},
ISSN = {00911798},
CODEN = {APBYAE},
MRCLASS = {60J80 (60G57 60J55 60J65)},
MRNUMBER = {93b:60195},
MRREVIEWER = {Klaus Fleischmann},
ZBLNUMBER = {0753.60078},
} 
[LixLy2006] X. Li and T. J. Lyons, "Smoothness of Itô maps and diffusion processes on path spaces. I," Ann. Sci. École Norm. Sup., vol. 39, iss. 4, pp. 649677, 2006.
@article {LixLy2006, MRKEY = {2290140},
AUTHOR = {Li, XiangDong and Lyons, Terry J.},
TITLE = {Smoothness of {I}tô maps and diffusion processes on path spaces. {I}},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {39},
YEAR = {2006},
NUMBER = {4},
PAGES = {649677},
ISSN = {00129593},
CODEN = {ASENAH},
MRCLASS = {60G17 (58J65 60J65)},
MRNUMBER = {2008a:60096},
MRREVIEWER = {Isamu D{ô}ku},
DOI = {10.1016/j.ansens.2006.07.001},
ZBLNUMBER = {1127.60033},
} 
[lyons] T. J. Lyons and Z. Qian, "System control and rough paths," in Oxford Math. Monogr., U.K.: Oxford University Press, 2002.
@incollection {lyons,
author={Lyons, Terry J. and Qian, Z.},
TITLE={System control and rough paths},
BOOKTITLE={Oxford Math. Monogr.},
PUBLISHER={Oxford University Press},
ADDRESS={U.K.},
YEAR={2002},
MRNUMBER={2005f:93001},
ZBLNUMBER={1029.93001},
} 
[LySiDoob] T. J. Lyons and N. Sidorova, "On the radius of convergence of the logarithmic signature," Illinois J. Math., vol. 50, iss. 14, pp. 763790, 2006.
@article {LySiDoob, MRKEY = {2247845},
AUTHOR = {Lyons, Terry J. and Sidorova, Nadia},
TITLE = {On the radius of convergence of the logarithmic signature},
JOURNAL = {Illinois J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {50},
YEAR = {2006},
NUMBER = {14},
PAGES = {763790},
ISSN = {00192082},
CODEN = {IJMTAW},
MRCLASS = {60H10 (34A34 34F05 93C35)},
MRNUMBER = {2007m:60165},
MRREVIEWER = {Habib Ouerdiane},
ZBLNUMBER = {1103.60060},
} 
[LyStFlour] T. J. Lyons, M. Caruana, and T. Lévy, Differential Equations Driven by Rough Paths, New York: SpringerVerlag, 2007.
@book {LyStFlour, MRKEY = {2314753},
AUTHOR = {Lyons, Terry J. and Caruana, Michael and L{é}vy, Thierry},
TITLE = {Differential Equations Driven by Rough Paths},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1908},
PUBLISHER = {SpringerVerlag},
ADDRESS = {New York},
YEAR = {2007},
PAGES = {xviii+109},
ISBN = {9783540712848; 3540712844},
MRCLASS = {60H10 (6002 60H07)},
MRNUMBER = {2009c:60156},
MRREVIEWER = {Bohdan Maslowski},
ZBLNUMBER = {05161505},
} 
[ohtsuka] M. Ohtsuka, "Dirichlet problems on Riemann surfaces and conformal mappings," Nagoya Math. J., vol. 3, pp. 91137, 1951.
@article {ohtsuka, MRKEY = {0045830},
AUTHOR = {Ohtsuka, Makoto},
TITLE = {Dirichlet problems on {R}iemann surfaces and conformal mappings},
JOURNAL = {Nagoya Math. J.},
FJOURNAL = {Nagoya Mathematical Journal},
VOLUME = {3},
YEAR = {1951},
PAGES = {91137},
ISSN = {00277630},
MRCLASS = {30.0X},
MRNUMBER = {13,642f},
MRREVIEWER = {R. de Possel},
URL = {http://projecteuclid.org/getRecord?id=euclid.nmj/1118799225},
ZBLNUMBER = {0043.30004},
} 
[Pis] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, Providence, RI: Amer. Math. Soc., 1986.
@book {Pis, MRKEY = {829919},
AUTHOR = {Pisier, Gilles},
TITLE = {Factorization of Linear Operators and Geometry of {B}anach Spaces},
SERIES = {CBMS Regional Conference Series in Mathematics},
NUMBER = {60},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS={Providence, RI},
YEAR = {1986},
PAGES = {x+154},
ISBN = {0821807102},
MRCLASS = {47A68 (46B20 47B10)},
MRNUMBER = {88a:47020},
MRREVIEWER = {Bernd Rosenberger},
ZBLNUMBER = {0588.46010},
} 
[Sh] G. R. Shorack, "Convergence of reduced empirical and quantile processes with application to functions of order statistics in the nonI.I.D. case," Ann. Statist., vol. 1, pp. 146152, 1973.
@article {Sh, MRKEY = {0336776},
AUTHOR = {Shorack, Galen R.},
TITLE = {Convergence of reduced empirical and quantile processes with application to functions of order statistics in the non{I}.{I}.{D}. case},
JOURNAL = {Ann. Statist.},
FJOURNAL = {The Annals of Statistics},
VOLUME = {1},
YEAR = {1973},
PAGES = {146152},
ISSN = {00905364},
MRCLASS = {60B10 (62G30)},
MRNUMBER = {49 \#1549},
MRREVIEWER = {G. E. Noether},
ZBLNUMBER={0255.62044},
}