Celebratio Mathematica

J. S. Birman and C. Series: “An algorithm for simple curves on surfaces,” J. London Math. Soc. (2) 29 : 2 (1984), pp. 331–342. MR 744104 Zbl 0507.57006 article

Let \( M \) be a compact orientable surface with non-empty boundary and with \( \chi(M) < 0 \), and let \( \Gamma = \pi_1 M \). Let \( \hat{C} \) be the free homotopy class of a closed loop on \( M \) and let \( W = W(\hat{C}) \) be a word in a fixed set of generators \( \overline{\Gamma} \) which represents \( \hat{C} \). In this paper we give an algorithm to decide, starting with \( W \), whether \( \hat{C} \) has a simple representative, that is a representative without self-intersections. Such a word will be said to be simple. As an application, we begin a study of simple words in \( \overline{\Gamma} \). Our results also apply to infinite geodesics on \( M \), corresponding to biinfinite words in \( \overline{\Gamma} \), where now we ask which finite blocks appear in such a word when the corresponding infinite homotopy class has a simple representative.

J. S. Birman and C. Series: “Geodesics with bounded intersection number on surfaces are sparsely distributed,” Topology 24 : 2 (1985), pp. 217–225. MR 793185 Zbl 0568.57006 article

Let \( M \) be a surface of negative Euler characteristic, possibly with boundary, which is either compact or obtained from a compact surface by removing a finite set of points. Let \( D \) be the Poincaré disc. Choose any representation of \( M \) as \( U/\Gamma \), where \( U \subseteq D \) is the universal covering space of \( M \) and \( \Gamma \subset \operatorname{Isom}(D) \). Then the Poincaré metric on \( D \) induces a metric of constant negative curvature on \( M \) and geodesics in \( U \) project to geodesics on \( M \). A geodesic on \( M \) is said to be complete if it is either closed and smooth, or open and of infinite length in both directions. complete geodesics coincide with those which never intersect \( \partial M \). Note that if \( M \) is obtained from a compact surface by removing a finite number of points to form cusps then a complete open geodesic on \( M \) might tend toward infinity along a cusp. In this paper we study the family \( G_k \) of complete geodesics which have at most \( k \) transversal self-intersections, \( k\geq 0 \).

J. S. Birman and C. Series: “Geodesics with multiple self-intersections and symmetries on Riemann surfaces,” pp. 3–11 in Low-dimensional topology and Kleinian groups (Coventry and Durham, UK, 1984). Edited by D. B. A. Epstein. London Mathematical Society Lecture Note Series 112. Cambridge University Press, 1986. MR 903857 Zbl 0618.57001 incollection

David Bernard Alper Epstein	Related
Caroline Mary Series	Related

J. S. Birman and C. Series: “Dehn’s algorithm revisited, with applications to simple curves on surfaces,” pp. 451–478 in Combinatorial group theory and topology (Alta Lodge, UT, 15–18 July 1984). Edited by S. M. Gersten and J. R. Stallings. Annals of Mathematics Studies 111. Princeton University Press, 1987. MR 895628 Zbl 0624.20033 incollection

John Robert Stallings, Jr.	Related
Stephen M. Gersten	Related
Caroline Mary Series	Related

J. S. Birman and C. Series: “Algebraic linearity for an automorphism of a surface group,” J. Pure Appl. Algebra 52 : 3 (July 1988), pp. 227–275. MR 952081 Zbl 0646.57007 article

Let \( M \) be a compact surface, \( \chi(M) < 0 \), and let \( \Gamma = \pi_1M \). Let \( S(M) \) be the set of isotopy classes of multiple simple loops on \( M \). Each \( \Lambda \in S(M) \) determines a family of cyclic words \( W = W(\Lambda) \), with associated coinitial graph \( \tau \). The finite set of coinitial graphs, obtained as \( \Lambda \) ranges over \( S(M) \), is interpreted as a set of ‘\( \pi_1 \)-train tracks’ on \( M \). The linearity theorem asserts that if a topologically induced automorphism \( \phi \) of \( \Gamma \) maps the set of weights \( W \) supported on \( \tau \) to a set supported on \( \tau^{\prime} \), then, with appropriate restrictions, the action is linear on the positive linear span of the \( W \)’s.

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Joan S. Birman

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