What the study found
The study calculates the index of a family of critical points of the systole function on Teichmüller space. It also gives an example of a minimal filling set of simple closed geodesics in genus 5 with cardinality 8, which the authors say are presumably realised as systoles.
Why the authors say this matters
The authors say these examples matter because their existence implies strata in the Thurston spine where systoles do not determine a basis for the surface homology. The findings also relate to the dimension of the Thurston spine near a critical point.
What the researchers tested
The researchers studied critical points of the systole function and related minimal filling subsets of systoles at those points. They examined the smallest and largest cardinalities of such minimal filling sets for tessellations of hyperbolic surfaces by regular, right-angled m-gons for m in {5, 6, 7}.
What worked and what didn't
The index calculation was completed for a family of critical points with the noted pathological feature. The study found a genus 5 example of a minimal filling set with 8 simple closed geodesics, and more generally determined the smallest and largest cardinalities for the tessellation cases considered. The abstract does not report any failed tests or negative results beyond noting that earlier index calculations had been impossible for the previously known examples.
What to keep in mind
The abstract does not give the numerical index values in the summary provided here. It also describes the genus 5 geodesic example as presumably realised as systoles, so that part is not stated as fully confirmed in the abstract.
Key points
- The study calculates the index of a family of critical points of the systole function on Teichmüller space.
- It gives a genus 5 example of a minimal filling set with 8 simple closed geodesics.
- The authors say these examples imply strata in the Thurston spine where systoles do not determine a homology basis.
- The researchers determined smallest and largest cardinalities for minimal filling sets in tessellations by regular, right-angled m-gons for m = 5, 6, 7.
Disclosure
- Research title:
- Small-genus examples of critical systole index calculated
- Image credit:
- Photo by WikiImages on Pixabay
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