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in Γ. The prime geodesics are then those geodesics that trace out their image exactly once — algebraically, they correspond to primitive hyperbolic conjugacy classes, that is, conjugacy classes {γ} such that γ cannot be written as a nontrivial power of another element of Γ.
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There are other similarities to number theory — error estimates are improved upon, in much the same way that error estimates of the prime number theorem are improved upon. Also, there is a
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There are then 3 types of transformation: hyperbolic, elliptic, and parabolic. (The loxodromic transformations are not present because we are working with
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This geodesic is closed because 2 points which are in the same orbit under the action of Γ project to the same point on the quotient, by definition.
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that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an
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In number theory, various "prime geodesic theorems" have been proved which are very similar in spirit to the
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The importance of prime geodesics comes from their relationship to other branches of mathematics, especially
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of a surface was in terms of simple closed curves. Closed geodesics have been instrumental in studying the
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proved a similar result for surfaces of variable negative curvature, while in his 1980 Ph.D. thesis,
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Algebraically, prime geodesics can be lifted to higher surfaces in much the same way that
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themselves. These applications often overlap among several different research fields.
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Closed geodesics have been used to study
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which are helpful in understanding prime geodesics.
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459:Splitting of prime ideals in Galois extensions
273:. This is a hyperbolic surface, in fact, a
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114:Learn how and when to remove this message
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430:which is formally similar to the usual
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407:). This result is usually credited to
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434:and shares many of its properties.
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231:, so Γ is a group of isometries of
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315:It can be shown that this gives a
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217:linear fractional transformation
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319:between closed geodesics on Γ\
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449:can be split (factored) in a
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421:Chebotarev's density theorem
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387:. To be specific, we let π(
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360:In dynamical systems, the
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219:. Each element of PSL(2,
182:Poincaré half-plane model
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419:proved an analogue of
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432:Riemann zeta function
428:Selberg zeta function
223:) in fact defines an
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385:prime number theorem
164:Technical background
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44:improve this article
571:Hyperbolic geometry
511:Modular group Gamma
297:called the axis of
189:hyperbolic geometry
170:hyperbolic geometry
494:Teichmüller spaces
461:for more details.
317:1-1 correspondence
281:of Γ determines a
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325:conjugacy classes
258:Now consider the
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187:of 2-dimensional
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546:Riemann surfaces
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413:Grigory Margulis
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260:quotient surface
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104:December 2009
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29:This article
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531:Zoll surface
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455:Covering map
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409:Atle Selberg
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42:Please help
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479:eigenvalues
128:mathematics
540:Categories
191:. Given a
136:hyperbolic
74:newspapers
486:operators
483:Laplacian
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31:does not
500:See also
366:periodic
225:isometry
471:Riemann
441:in the
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52:removed
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453:. See
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