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It is well-known that the fundamental group of a twice-punctured torus is a free group of rank three.

I see that there is no one-to-one correspondence between the homotopy classes of essential simple loops on twice-punctured torus and the conjugacy classes of primitive elements in a free group of rank three.

Do we know which primitive elements in a free group of rank three represent simple loops on a twice-punctured torus?

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3 Answers 3

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A classification of primitive elements in a free group of rank greater than two is a hard problem, and there is no really satisfactory classification known. I am pretty sure this paper of Shpilrain is pretty close to the last word. As for elements representing simple closed curves, this is also not easy, and the best results are algorithmic results of D. Chillingworth (MR0248819 (40 #2069) Chillingworth, D. R. J. Simple closed curves on surfaces. Bull. London Math. Soc. 1 1969 310–314), which were essentially replicated by Birman-Series (MR0744104 (85m:57002) Birman, Joan S.(1-CLMB); Series, Caroline(4-WARW) An algorithm for simple curves on surfaces. J. London Math. Soc. (2) 29 (1984), no. 2, 331–342. ) and Cohen-Lustig (MR0895629 (88m:57016) Cohen, Marshall(1-CRNL); Lustig, Martin(1-MIT) Paths of geodesics and geometric intersection numbers. I. Combinatorial group theory and topology (Alta, Utah, 1984), 479–500, Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, NJ, 1987. 57N05 (11F06) )

EDIT One relevant remark: The number of conjugacy classes of simple closed curves grows polynomially (the rates of growth were determined by I. Rivin in "Simple curves on surfaces", and corresponding asymptotic results were obtained by M. Mirzakhani. The number of conjugacy classes of primitive elements, however, grows exponentially (easy construction for $F_3:$ take $x_3$ times any word in $x_1, x_2$). This is why things become much harder past $F_2.$

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Many thanks, Professor Rivin!! Your answer helped me a lot. –  scarlet Nov 27 '11 at 17:13

There are numerous algorithms to decide this question. At bottom all of these are based on the "monogon" and "bigon" condition: If $\alpha$ is a closed loop on a surface then we can homotope $\alpha$ to realize its minimal self-intersection number by looking for and then removing mongons and bigons. For an example of such a paper, see Chillingworth's "Winding numbers on surfaces. II". For a related discussion, see section 1.2.4 of the "Primer on mapping class groups" by Farb and Margalit.

There is also a more geometric criterion having to do with "linking at infinity." See section 8.2.4 of the Primer.

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Is there a more elementary reference for the "monogon" and "bigon" condition for a single closed curve? The one in "Primer on mapping class groups" is a proof about pairs of simple closed curves. I'm working on an application that only consider this problem for the plane with holes, so it be nice to just learn the special case without going too deep into the theory. –  Chao Xu Oct 5 '13 at 0:58
@ChaoXu - I spent a while thinking about this and finally had the clever idea of asking Dylan Thurston. He pointed to his recent paper (arxiv.org/abs/1310.1959). See in particular Section 2 and the references to the papers of Hass-Scott. –  Sam Nead Oct 25 '13 at 2:31

There are 3 types of simple closed curves on a twice punctured torus.

The first type are two isotopy classes of curves which are isotopic to the two boundary components. These are primitive and separating.

The second type is separating curves which cut the torus into a once-punctured torus and a pair of pants. These curves are not primitive, since they are commutators (although I assume this is not what you mean when you observe there is no 1-1 correspondence between primitive curves and simple closed curves).

The third type is non-separating curves which are primitive. These curves have the property that they are sent to their inverse (when keeping track of orientation) under the elliptic involution (which exchanges the two boundary components). So this property is certainly a restriction: a primitive curve conjugate to a simple closed curve which is not isotopic to a boundary component must be sent to its inverse under the elliptic involution. I think this symmetry condition is not sufficient, since the subgroup of $Aut(F_3)$ commuting with this involution is larger than the mapping torus of the twice-punctured torus.

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This might or might not tell you which simple curves are primitive, but not which primitive curves are simple, alas. –  Igor Rivin Nov 28 '11 at 10:02

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