The A-model partition function is a topological invariant of any hyperbolic surface. So, what is this invariant? I’m not sure if there is a way to normalize these since the usual perscription Z(S2)=1 doesn’t work here, but I figure there’s a way to do it anyway. EDIT: It seems like the invariant should just be … Read more
UPDATE – Feb. 9, 2017: The original title of this post was “The $\text{isometry}^+$ group of hyperbolic $n$-space as $\mathrm{PSL}_2$ of a Clifford group.” The original question, which appears below, did not receive answers, but I discovered in the meantime that the confusion arrises from conflicting notation in the literature. So I’ve explained that in … Read more
A few years ago, Roberto Frigerio asked for a reference for a geometric property of horospheres (Reference for the geometry of horospheres), namely exponential decay of the projection onto a horosphere. My question is: does a similar exponential decay still holds in a Gromov hyperbolic space ? The precise statement would be something like: Let … Read more
I apologize in advance if this is elementary; I have very little experience with hyperbolic manifolds, but I’m using hyperbolic manifolds for part of a current project. Given a discrete torsion-free subgroup of isometries acting on hyperbolic space Γ↪Isom(Hn), then Γ also acts on the boundary sphere Sn−1 of hyperbolic space. As I understand it, … Read more
Two easy questions: Given a metric space $(Z,d_Z)$, let $X = \text{Con}_h(Z) = Z \times ]0,\infty[$ and for $(z,i), (z’,i’) \in \text{Con}_h(Z)$ define a metric on $X$ as follows: $d((z,i), (z’,i’)) = d_{\mathbb{H}^2}((0,i), (d_Z(z,z’),i’) = \text{arcosh}(1+\frac{d_Z(z,z’)^2 + (i-i’)^2}{2ii’})$. Question: If $Z$ was a $\text{CAT}(0)$ space, is then $X$ $\text{CAT}(-1)$? Question 2: If not, is there … Read more
Let X be a complete finite-volume orientable hyperbolic 3-manifold, and let Γ be a Kleinian representation of π1(X). Let KΓ:=Q({tr∣γ∈Γ}), i.e. its trace field. Let Γ(2):=⟨γ2∣γ∈Γ⟩, the group generated by squares in Γ and let kΓ=KΓ(2), i.e. its invariant trace field. Then these are number fields, KΓ is a manifold invariant and kΓ is a … Read more
Is there an Intercept theorem (from Thales, but don’t mistake it with the Thales theorem in a circle) in hyperbolic geometry? Euclidean Intercept Theorem: Let S,A,B,C,D be 5 points, such that SA, SC, AC are respectively parallel to SB, SD, BD; and SB > SA, SD > SC. Then, we have SB/SA = SD/SC = … Read more
By examining numerous examples I have become quite convinced that the following statement is true. Let x and y be distinct points in the hyperbolic plane H. Let γ be the geodesic passing through x and y and let vx and vy be the unit tangent vectors to γ at x and y respectively. We … Read more
If Σ is a smooth embedded closed hypersurface in Rn with (normalized) mean curvature H≤1 (the mean curvature of the unit sphere), then its ((n−1)-dimensional) area is at least the area of the unit sphere in Rn. This is proved in Chavel’s book “Isoperimetric inequalities” Theorem II.1.3. The proof uses the idea that by restricting … Read more
The boundary of an $r$-neighborhood of the convex core of hyperbolic $n$-manifold is smooth, by page 73 of Hyperbolic Manifolds and Kleinian Groups. The authors does not provide a proof for this fact. This does not look very obvious as the $r$-neighborhood of a convex set in a Riemannian manifold may not be smooth. One … Read more
