On the number and location of short geodesics in moduli space

Leininger, Christopher J.; Margalit, Dan

doi:10.1112/jtopol/jts025

Cited by 9 publications

(10 citation statements)

References 30 publications

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“…We do not know the answer, but the examples from Theorem 1.1 provide a place to start. Other works studying the shape of moduli space in relation to the systole include [3,16,19,21,23,24,39,41,46].…”

Section: Motivationmentioning

confidence: 99%

Local maxima of the systole function

Bourque

Rafi

2021

J. Eur. Math. Soc.

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We construct a sequence of closed hyperbolic surfaces that are local maxima for the systole function in their respective moduli spaces. Their systole is arbitrarily large and the number of examples grows rapidly with the genus. More precisely, for every n 3 there is some positive number L n (growing roughly linearly in n) such that the number of local maxima of the systole function in genus g with systole equal to L n grows super-exponentially in g along an arithmetic sequence of step size n. Many of these surfaces have no orientation-preserving isometries other than the identity and are the first examples of local maxima with this property.

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Section: Motivationmentioning

confidence: 99%

Local maxima of the systole function

Bourque

Rafi

2021

J. Eur. Math. Soc.

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“…For the remaining cases, we may take the following, again as suggested by Jürgen Müller: for g = 5, 13, 25, 41 : G g = (C 4 × C 2 ) C 4 = SmallGroup (32,2) for g = 7, 16, 34 : G g = C 9 C 3 = SmallGroup (27,4) for g = 23 : G g = SL 2 (3) C 4 = SmallGroup(96,66) Again, we can appeal to Breuer's catalog to see that G g is a subgroup of Mod(S g ), and the same character theory argument shows that these groups are not subgroups of SO (5), nor do they have a subgroup of index 2 in SO (4). By the theorem of Mecchia-Zimmermann, we then have than these G g do not lie in Diff + (S 4 ).…”

Section: Homomorphisms To Homeomorphism Groups Of Spheresmentioning

confidence: 99%

“…Theorem 5 (Theorem 1.1, [27]). For g ≥ 3, every nontrivial periodic mapping class that is not a hyperelliptic involution normally generates Mod(S g ).…”

Section: Introductionmentioning

confidence: 99%

Constraining mapping class group homomorphisms using finite subgroups

Chen¹,

Lanier²

2021

Preprint

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We classify homomorphisms from mapping class groups by using finite subgroups. First, we give a new proof of a result of Aramayona-Souto that homomorphisms between mapping class groups of closed surfaces are trivial for a range of genera. Second, we show that only finitely many mapping class groups of closed surfaces have non-trivial homomorphisms into Homeo(S n ) for any n. We also prove that every homomorphism from Mod(Sg) to Homeo(S 2 ) or Homeo(S 3 ) is trivial if g ≥ 3, extending a result of Franks-Handel.

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“…Example of a diamond. The following example of a pair of fibers in a 3-manifold fibering over the circle is taken from [1], Lemma 5.1.…”

Section: 2mentioning

confidence: 99%

“…It is argued in [1], Lemma 5.1, that the homology class 4, is therefore also a fiber, with pseudo-Anosov monodromy. The monodromy of the fiber S 2 can not be in the Torelli group, because the genus of S 2 is larger than the genus of S 1 .…”

Section: 2mentioning

confidence: 99%