# Teichmüller space

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A metric space $(M_g,d)$ with as points abstract Riemann surfaces (that is, classes $\bar X$ of conformally-equivalent Riemann surfaces $X$ of genus $g$ (cf. Riemann surfaces, conformal classes of) with singled out equivalent (with respect to the identity mapping) systems $\Sigma$ of generators of the fundamental group $\pi_1(X)$, and in which the distance $d$ between $\bar X$ and $\bar X'$ is equal to $\log K$, where the constant $K$ is the dilatation of the Teichmüller mapping (of the quasi-conformal mapping $\bar X \rightarrow \bar X'$ giving the smallest maximum dilatation among all such mappings). Introduced by O. Teichmüller [1].

#### References

 [1] O. Teichmüller, "Extremale quasikonforme Abbildungen und quadratische Differentialen" Abhandl. Preuss. Akad. Wissenschaft. Math.-Nat. Kl. , 22 (1939) pp. 3–197 [2a] L. Bers, "Quasi-conformal mappings and Teichmüller's theorem" R. Nevanlinna (ed.) et al. (ed.) , Analytic functions , Princeton Univ. Press (1960) pp. 89–119 [2b] L.V. Ahlfors, "The complex analytic structure of the space of Riemann surfaces" R. Nevanlinna (ed.) et al. (ed.) , Analytic functions , Princeton Univ. Press (1960) pp. 45–66 [2c] L. Bers, "Spaces of Riemann surfaces" , Proc. Intern. Congress Mathematicians, Edinburgh 1958 , Cambridge Univ. Press (1959) pp. 349–361 [2d] L. Bers, "Simultaneous uniformization" Bull. Amer. Math. Soc. , 66 (1960) pp. 94–97 [2e] L. Bers, "Holomorphic differentials as functions of moduli" Bull. Amer. Math. Soc. , 67 (1961) pp. 206–210 [2f] L. Ahlfors, "On quasiconformal mappings" J. d'Anal. Math. , 3 (1954) pp. 1–58; 207–208 [3] S.L. Krushkal, "Quasi-conformal mappings and Riemann surfaces" , Halsted (1979) (Translated from Russian)