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38
votes
6answers
3k views

What do Weierstrass points look like?

As somebody who mostly works with smooth, real manifolds, I've always been a little uncomfortable with Weierstrass points. Smooth manifolds are totally homogeneous, but in the complex category you ...
24
votes
2answers
716 views

chromatic number of the hyperbolic plane

A notorious problem in combinatorics is the following: If we color $\mathbb{R}^2$ so that no pair of points at unit distance get the same color, what is the fewest number of colors required? This ...
19
votes
2answers
256 views

Hyperbolic exceptional fillings of cusped hyperbolic 3-manifolds

Thurston's Hyperbolic Dehn Surgery Theorem says that all but finitely many fillings of a cusp of a hyperbolic 3-manifold result in hyperbolic manifolds that are deformations of the original manifold. ...
18
votes
5answers
2k views

〈x,y : x^p = y^p = (xy)^p = 1〉

Let $p$ be an odd prime and $G := \langle x,y : x^p = y^p = (xy)^p = 1 \rangle$. I want to show that $G$ is infinite and wonder if there is a good way to prove this. I'm familiar with the basics of ...
17
votes
3answers
1k views

Is there a volume conjecture for closed 3-manifolds?

A typical statement of the volume conjecture, for instance in Murakami's survey 1002.0126, is Conjecture: For $K$ a knot in $S^3$, the N-th colored Jones polynomials are related to the volume of ...
17
votes
2answers
804 views

How does hyperbolicity of space time affect our lives?

My main research has been in hyperbolic geometry and geometric group theory. I always thought that the only real "application" of my work was that the universe is a 3-manifold. But recently I found ...
17
votes
2answers
478 views

Is every elementary absolute geometry Euclidean or hyperbolic?

Absolute geometry is any one that satisfies Hilbert's axioms of plane geometry without the axiom of parallels. It is well-known that it is either the Euclidean or a hyperbolic plane. For an elementary ...
17
votes
3answers
733 views

Hyperbolic Coxeter polytopes and Del-Pezzo surfaces

Added. In the following link there is a proof of the observation made in this question: http://dl.dropbox.com/u/5546138/DelpezzoCoxeter.pdf I would like to find a reference for a beautiful ...
16
votes
2answers
684 views

Non-residually finite matrix groups

By Malcev's theorem, every finitely generated linear group is residually finite (RF). On the other hand, say, the group of rational numbers is linear, but is not residually finite. Thus, one has to ...
16
votes
2answers
2k views

A question about the proof of Mostow rigidity

I have recently been studying a proof of Mostow rigidity (along the lines of Mostow's original argument), and I'm left a little confused about something. We start with an isomorphism $\alpha: \Gamma ...
16
votes
2answers
834 views

Maximum of a function of one variable

Let $D$ be a circular quadrilateral (that is a Jordan region whose boundary consists of 4 arcs of circles all orthogonal to the unit circle) whose interior angles are all equal to 0, the vertices lie ...
15
votes
6answers
998 views

Tetrahedra with prescribed face angles

I am looking for an analogue for the following 2 dimensional fact: Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is ...
15
votes
3answers
736 views

The number of cusps of higher-dimensional hyperbolic manifolds

Suppose $n$ is an integer greater than 3. Sometimes ago I heard somewhere that it is still not known if there exist complete finite-volume hyperbolic $n$-manifolds having exactly one cusp. Could ...
15
votes
0answers
399 views

Actions on ℍⁿ generated by torsion elements

Let $n$ be a large integer. I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order. Or equivalently, ...
14
votes
5answers
2k views

can you fool SnapPea?

A while back I thought I had some simple knots that "fooled" SnapPea. But I no longer remember those examples, if I ever had them to begin with. What I'm looking for is a non-hyperbolizable knot ...
14
votes
1answer
559 views

Why is there a unique hyperbolic simplex of largest area?

Why is there a unqiue ideal $n$-simplex in $\mathbb H^n$ with largest volume for $n\geq 3$? For $n=3$, this is a standard calculation, and for larger dimensions is much harder (see Haagerup and ...
14
votes
1answer
654 views

Locus of equal area hyperbolic triangles

Henry Segerman and I recently considered the following question: Given a fixed area $A < \pi$ and two fixed points in the upper half-plane model for hyperbolic $2$-space, what is the locus of ...
13
votes
2answers
381 views

For which surfaces is Penner's conjecture known to be true?

Robert Penner has proven that, if $A=\{a_1,\dots, a_n\}$ and $B=\{b_1,\dots, b_m\}$ are multicurves in a surface $S$ that together fill $S$, then any product of positive powers of Dehn twists along ...
13
votes
3answers
429 views

Failure of Mostow rigidity in dim. 2

I am trying to understand why Mostow rigidity fails in dimension 2. More concretely, I have the following question: (1) What is an example of a quasiisometry $f$ of the hyperbolic plane $\mathbb H^2$ ...
13
votes
1answer
1k views

Detecting a cover of the figure-8 knot complement

I have a specific concrete example of a complete, finite volume, cusped hyperbolic 3-manifold $M$, and I am trying to determine whether or not $M$ is a cover of the figure-8 knot complement (call this ...
12
votes
3answers
642 views

F→E→B bundle with B,E,F hyperbolic: possible?

It would be interesting to me obtain an answer to the following easy to state question: Does there exist a (smooth) fibre bundle $\pi\colon E\rightarrow B$ with typical fibre $F$ such that $E$, $B$ ...
12
votes
1answer
369 views

Does a compact negatively curved manfiold of dimension 4 admit a cover of finite degree?

A $3$-dimensional compact manifold of negative sectional curvature admits (by geometrisation?) a metric of curvature $-1$, and so its fundamental group has subgroups of finite index. I wonder if an ...
12
votes
1answer
484 views

Comparing layered triangulations of 3-manifolds which fiber over the circle.

I am sorry but I am reposting this question because I wasn't logged in when I first asked it. Ian Agol has produced a method to build an ideal layered triangulation of a hyperbolic 3-manifold which ...
12
votes
1answer
699 views

Gromov's Hyperbolicity and Positive Cheeger Constant in Planar Graphs

It is known that for metric graphs the concepts of Gromov's hyperbolicity and strictly positive Cheeger constant are related. Let us first recall the definition of the Cheeger constant. Let $G$ be a ...
12
votes
1answer
418 views

Distortion of malnormal subgroup of hyperbolic groups

Let $G$ be a countable, Gromov-hyperbolic group. We say that $H$ is hyperbolically embedded in $G$ if $G$ is relatively hyperbolic to {$H$} (in the strong sense). This definition is due to Osin. A ...
11
votes
3answers
272 views

Origin of number theoretic invariants associated to hyperbolic 3-manifolds

I've been studying number theoretic methods of classifying hyperbolic 3-manifolds for over a year now. In particular, there is are the trace field, invariant trace field, quaternion algebra, and ...
11
votes
2answers
603 views

The fundamental group of a closed surface without classification of surfaces?

The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation $$ \langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle. $$ The proof I know ...
11
votes
1answer
280 views

Borel-Serre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$

Let $\mathbb{H}^3$ be the three-dimensional hyperbolic space. Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Then $SL_2(\mathcal{O}_K)$ acts on $\mathbb{H}^3$ ...
11
votes
1answer
204 views

Rank and hyperbolic volume

Suppose $M$ is a hyperbolic $3$-manifold whose fundamental group has rank $r.$ What is the best (lower) bound on the volume of $M?$ Similar question for rank of $H_1.$ There is a bunch of papers of ...
11
votes
2answers
184 views

Is $\mathbb{H}^n$ quasi-isometric to a leaf of a codimension 1 foliation of a compact manifold?

If we extend the action of $\pi_1(\Sigma_g), g\geq 2,$ from $\mathbb{H}^2$ to its boundary $\partial_{\infty}\mathbb{H}^2=S^1$, the surface bundle corresponding to this action of $\pi_1(\Sigma_g)$ on ...
11
votes
1answer
227 views

Hyperbolic Manifolds which fiber over the circle

If $N^2$ is a closed, orientable surface of genus at least $2$, and if $\phi$ is an (orientation-preserving) pseudo-Anosov mapping on $N$, then one can form the closed orientable 3-manifold $M^3$ by ...
10
votes
3answers
800 views

Flat SU(2) bundles over hyperbolic 3-manifolds

Can someone give me a non-trivial example of a flat SU(2)-connection over a compact orientable hyperbolic 3-manifold? The literature on such bundles over 3-manifolds is huge and my naive searches ...
10
votes
3answers
435 views

Torsion in cuspidal cohomology

Following Lemma 2.7 from Vogtmann's Rational Homology of Bianchi Groups, I want to define cuspidal cohomology as $$H_{\mathrm{cusp}}(M)=\frac{H_1(M)}{i_*(H_1(\partial M))}$$ where $i:\partial M\to M$ ...
10
votes
2answers
2k views

The work of Thurston

I seem to remember written or said somewhere that at some point Thurston decided to stop writing down his theorems in order not to repel mathematicians from his field (maybe this is not correct?). I ...
10
votes
2answers
754 views

Closed hyperbolic manifold with right-angled fundamental domain

What is an example (as simple as possible, please!) of a closed hyperbolic three-manifold with a right-angled polyhedron as fundamental domain? If we allow cusps then the Whitehead link or the ...
10
votes
2answers
379 views

Some mid-sized ¿hyperbolic? manifolds and SnapPea

I've recently come across some mid-sized 3-manifolds that I think are likely hyperbolic, but SnapPea has some trouble with them. This is related to my previous question can you fool SnapPea? but in ...
10
votes
3answers
591 views

Best known Margulis constants?

A Margulis number for a hyperbolic $n$-manifold $M=\mathbb{H}^n/\Gamma$ is a number $\epsilon>0$ such that for each $x\in\mathbb{H}^n$ the group generated by the elements in $\Gamma$ which move $x$ ...
10
votes
1answer
285 views

Examples of 3-manifolds with RFRS fundamental group

I'm wondering if anyone knows how to construct hyperbolic 3-manifolds whose fundamental group is RFRS. Clearly the recent work of Agol, Wise, etc. says that such manifolds are abundant, and in ...
10
votes
1answer
263 views

Representation varieties of 3-manifold groups in SL(n,C)

I am looking at the variety of representations of the fundamental group of a hyperbolic 3-manifold into $SL(n,C)$: $$Hom(\pi_1M, SL(n,{\mathbb C}))$$ It is known that volume and Chern-Simons ...
10
votes
1answer
493 views

Measure on the Boundary of a Hyperbolic Group

Let $\Gamma$ be a non-elementary Gromov's $\delta$-hyperbolic group. Let $B(1,n)$ be the set of elements at distance at most $n$ from the identity and let $\partial B(1,n)$ be the elements at distance ...
10
votes
1answer
1k views

Pythagorean Theorem for Right-Corner Hyperbolic Simplices?

My answer to the "Favorite Equations" question was the Pythagorean Theorem for Right-Corner Tetrahedra: Euclidean: $A^2+B^2+C^2=D^2$ Hyperbolic: ...
9
votes
6answers
704 views

How to smootly interpolate between möbius transformations?

If you have two Möbius transformations represented as: $f(z) = \frac{az + b}{cz + d}$ $g(z) = \frac{pz + q}{rz + s}$ where $a, b, c, d, p, q, r, s, z \in \mathbb{C}$ Is it possible to derive a ...
9
votes
4answers
612 views

Geometry of the space of circles in the Euclidean plane

We know that Mobius transformations, $z\to\frac{az+b}{cz+d}$, permutes circles and lines in the Euclidean plane, $(\mathbb{R}^2, dx^2 + dy^2 )$. It may even be possible to write an explicit formula ...
9
votes
4answers
436 views

When are those subgroups of $\mathrm{SL}(2, \mathbb{C})$ discrete?

Let $A = \pmatrix{1 & 0 \\ \alpha & 1} $ and $ B = \pmatrix{1 & 1 \\ 0 & 1}$, where $\alpha \in \mathbb{C}$ is a complex parameter. Now consider the family of representations ...
9
votes
2answers
971 views

Existence of finite index torsion free subgroups of hyperbolic groups

Question. Is it true that each infinite hyperbolic group has a torsion free subgroup of finite index? Are there counterexamples, or positive results for some large subclasses of hyperbolic groups? ...
9
votes
2answers
596 views

How to see isometries of figure 8 knot complement

The figure 8 knot complement $M$ is the orientable double cover of the Gieseking manifold, which implies that $M$ has a fixed-point free involution. If we think of $M$ with its hyperbolic metric, this ...
9
votes
3answers
623 views

Primitive elements in a free group of rank three

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 ...
9
votes
2answers
297 views

Heegaard genera of arithmetic 3-manifolds

UPDATE: Because I was hoping that state the question as concisely as possible, the original post did not include a precise definition of arithmetic 3-manifold only a reference to Maclachlan and ...
9
votes
4answers
771 views

Abel's equation for the dilog

Abel's identity for the dilogarithm (see the wikipedia page about polylogarithms) plays a role in web geometry as it is one of the abelian relations of the first example of exceptional web (Bol's ...
9
votes
1answer
300 views

Topological rigidity for negatively curved manifolds?

I was wondering if two compact oriented manifold carrying a Riemannian metric with negative sectional curvature, whose fundamental groups are isomorphic, are necessarily diffeomorphic (or ...