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2
votes
0answers
77 views
Geometrization & JSJ decomposition with boundary
Is there any paper where I can find a good explanation of the JSJ decomposition, the geometrization theorem and the relations between them when the manifold has nonempty (and non necessarily toroidal) ...
0
votes
1answer
80 views
A question on Cayley graphs and hyperbolic 3-manifolds
There are two hyperbolic closed 3-manifolds, but I don't know whether they are homeomorphic or not. The only thing I know is that the Cayley graphs of their fundamental groups are quasi-isometric.
...
5
votes
1answer
163 views
Can one use the continuity method to show that the two dimensional hyperbolic space can be immersed in five dimensional Euclidean space?
First of all, I must clarify at the outset that I am simply asking if there is an alternative way to solve an already known problem. It is known that the answer to my question is yes. The problem is ...
4
votes
0answers
81 views
Haken manifolds and characterising sutured manifold hierarchies
In Gabai's paper (Knot Theory and Manifolds Lecture Notes in Mathematics Volume 1144, 1985, pp 14-17 An internal hierarchy for 3-manifolds) he considers sutured manifold decompositions of Haken ...
2
votes
1answer
73 views
The measure on the harmonic spectrum from Selberg trace formula
One can see the following two equations,
Theorem 6.1 (Selberg Trace formula) on page 26 of these notes.
Equation 3.19 and 3.20 on page 11 of this paper.
I vaguely feel that these two are the ...
8
votes
1answer
99 views
Counterexamples to analogue of Cannon conjecture in higher dimensions
It is known that a group $G$ acts geometrically on $\mathbb{H}^2$ if and only if $G$ is word-hyperbolic and its boundary $\partial G$ is homeomorphic to $S^1$.
The analogous statement for ...
5
votes
1answer
219 views
Fixed points on boundary of hyperbolic group
Let G be a word-hyperbolic group with torsion and let ∂G be its boundary. Do there exist criteria that imply that all non-trivial finite order elements of G act fixed-point freely on ∂G?
3
votes
0answers
80 views
The distance between two farthest points on the Bolza surface?
The Bolza surface $M$ is the closed hyperbolic surface of genus $2$ that can be obtained by identifying the opposite sides of the regular octagon in $\mathbb{H}^2$.
What two points on $M$ are ...
6
votes
3answers
344 views
Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$
Let $X$ be a smooth projective complex algebraic variety of general type. Suppose that the (topological) fundamental group of $X$ is an infinite abelian group and that $\pi_2(X^{an})$ is finite.
What ...
4
votes
1answer
151 views
Andreev's Theorem and Thurston's hyperbolization theorem
I am attempting to get to grips with Thurston's hyperbolization theorem for Haken $3$--manifolds. In particular I was looking at the section related to gluing up along hierarchy surfaces in Otal, ...
1
vote
0answers
129 views
Easy explanation of non-abelianness of hyperbolic curves [migrated]
I'm looking for easy proofs (or just an easy proof) of the following statement:
Let X be a hyperbolic Riemann surface, i.e., $X$ is a Riemann surface and the universal covering of $X$ is the complex ...
5
votes
1answer
156 views
Standard (special) spines and hyperbolic structure on 3-manifolds
My question relates to constructing angled triangulations or hyperbolic triangulations for $3$--manifolds. Briefly, an angle triangulation can be considered as an assignment of a real number (called ...
6
votes
0answers
122 views
Spectral theory for Dirac Laplacian on a funnel
I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...
0
votes
0answers
51 views
Existence of special pants decompositions for non-elementary representations into PSL(2,R)
A Theorem by Gallo, Goldman and Porter states the following:
Let $S_g$ be a closed orientable surface of genus $g$ with fundamental group $\Gamma_g$, and fix a non-elementary representation ...
0
votes
0answers
49 views
Universal covering of a submanifold with boundary
Good afternoon,
I'm having a hard time trying to prove the following lemma :
Let $V$ be a hyperbolic compact orientable manifold, containing two totally geodesic com pact hypersurfaces $H$ and $G$ ...
8
votes
1answer
156 views
Dehn's algorithm for word problem for surface groups
For some $g \geq 2$, let $\Gamma_g$ be the fundamental group of a closed genus $g$ surface and let $S_g=\{a_1,b_1,\ldots,a_g,b_g\}$ be the usual generating set for $\Gamma_g$ satisfying the surface ...
2
votes
0answers
102 views
Discrete Isoperimetric Problem in the Hyperbolic Plane
Let $S_{g}$ be the closed orientable genus $g$ surface. I'm interested in studying a certain class $\Gamma$ of graphs on $S_{g}$ which fill (the complementary regions are simply connected), and which ...
8
votes
2answers
309 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 ...
3
votes
2answers
97 views
Comparing two Delaunay tessellations on a hyperbolic surface
Let $S$ be a closed hyperbolic surface (i.e. a compact Riemann surface of genus $\geq 2$) and let $P=\{p_1,\ldots,p_m\}$ be a non-empty finite subset of $m$ points in $S$. Let $\pi:\mathbb ...
5
votes
1answer
140 views
Topology of boundaries of hyperbolic groups
For many examples of word-hyperbolic groups which I have seen in the context of low-dimensional topology, the ideal boundary is either homeomorphic to a n-sphere for some n or a Cantor set. So, I was ...
6
votes
1answer
132 views
Heegaard genus of covers of cusped hyperbolic 3-manifold
Is there a cusped (non-compact) orientable hyperbolic 3-manifold $M$ which has Heegaard genus $g$, and has a finite-sheeted cover with Heegaard genus $<g$?
Moreover, if the cover is index $n$, then ...
5
votes
2answers
158 views
Comparing different layered structures for fibered 3-manifolds: example request.
Let's consider a fibering hyperbolic 3-manifold obtained as a mapping torus over some hyperbolic surface with pseudo-Anosov monodromy, and let's suppose that the surface is punctured at the singular ...
2
votes
1answer
147 views
Hyperbolic structures on once punctured tori
I've been working on a problem about billiards in ideal hyperbolic polygons and I was thinking about how the problem for ideal quadrilaterals relates to closed geodesics on once punctured tori.
My ...
12
votes
1answer
267 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 ...
0
votes
0answers
43 views
Geometric effects of removing elements of D2n generalizable?
So, if I start with a full Dihedral group D2n to represent a regular, ideal polygon in the hyperbolic plane, then I remove an element (and any subsequently necessary elements so that it is still a ...
5
votes
2answers
267 views
Negative sectionnal curvature and constant curvature
Good morning everyone,
I was wondering about the difference between manifolds carying a Riemanniann metric with negative sectionnal curvature and hyperbolic manifolds. I was told once "there are ...
4
votes
1answer
138 views
Fundamental group of an hyperbolic $4$-manifold
Good afternoon everyone,
I have a very general question about hyperbolic manifolds and their fundamental groups in high dimension(at least $4$). If the theory of surfaces and $3$-manifolds provide a ...
1
vote
2answers
465 views
Hyperbolic pair of pants.
Suppose $Y$ is a pair of pants with a hyperbolic structure and $\gamma_i; i = 1, 2, 3$ are the geodesic boundaries of length $l_i; i=1, 2, 3$ respectively. Now consider a essential simple arc $\sigma$ ...
2
votes
1answer
126 views
Purely parabolic Kleinian groups
What can be said about a discrete finitely generated subgroup $G$ of $PSL(2,\mathbb C)$ whose
nontrivial elements are parabolic? If $G$ is geometrically finite, one can show that $G$ must be ...
7
votes
1answer
345 views
Why isn't $\langle x,y,z|xyzx^{-1}y^{-1}z^{-1}\rangle$ a hyperbolic surface group?
The group mentioned in the title, $\langle x,y,z|xyzx^{-1}y^{-1}z^{-1}=1\rangle$, is in between the torus fundamental group $\langle x,y|xyx^{-1}y^{-1}=1\rangle$ and the two-holed torus fundamental ...
2
votes
2answers
152 views
Mid point with set square?
Is it possible to construct the midpoint of a segment in the hyperbolic plane
using the set square only?
With the set square one can
draw the line through the given two points and
drop the ...
1
vote
1answer
190 views
An action or two of $SL_2(\Bbb Z)$?
$SL_2(\Bbb Z)$ acts on ${\Bbb R}^2$ fixing set-wise ${\Bbb Z}^2$, so $SL_2(\Bbb Z)$ acts on
${\Bbb R}^2\setminus {\Bbb Z}^2$, and then on the universal covering space of ${\Bbb R}^2\setminus {\Bbb ...
3
votes
2answers
178 views
Do quasi convex hyperbolic subgroups remain quasi convex after adding redundant generators?
We know now that hyperbolic 3-manifolds virtually embed into right-angled Artin groups as quasiconvex subgroups. Also, quasiconvexity depends on the generating set.
I have been constructing a space ...
2
votes
1answer
167 views
Fixed point set of an isometric group action on an hyperbolic manifold
Good morning,
I'm trying to understand the following fact, that is stated in Gromov and Thurston's paper "Pinching constants for hyperbolic manifolds" :
Let $M$ be a (at least) 3-dimensional compact ...
1
vote
1answer
51 views
Fixed submanifolds of the sphere at infinity of $\mathbb{H}^n$
Good afternoon,
Take a submanifold $V$ of codimension $1$ of the sphere at infinity of $\mathbb{H}^n$ which is not the sphere at infinity of a totally geodesic hyperplane $\mathbb{H}^{n-1} \subset ...
7
votes
1answer
149 views
Is the group of integer points of ${\rm SO}(n,1)$ maximal?
That is, is it true that there does not exist a lattice in $G = {\rm SO}(n,1)$ which contains the group of integer points of $G$ as a proper subgroup (obviously then of finite index)? if such a ...
4
votes
3answers
202 views
injectivity radius of hyperbolic surface
Given a positive real number $l$. Does there exist a closed hyperbolic surface $X$ so that injectivity radius not less than $l$?
6
votes
2answers
290 views
Do different Dehn fillings produce homeomorphic 3-manifolds ?
Hi, everyone. I am interested in the dehn filling and Hyperbolic 3-manifold.
Suppose M be an orientable compact 3-manifold with one torus boundary and int(M) admit a
hyperbolic structure. ...
5
votes
1answer
192 views
When are isometry groups of hyperbolic 3-manifolds finite?
If $M$ is a finite volume hyperbolic 3-manifold, then its isometry group is finite. I believe this is also true for geometrically finite 3-manifolds. What is the most general condition on a hyperbolic ...
17
votes
2answers
707 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 ...
5
votes
0answers
250 views
A conjecture of Thurston and possibly Weeks too
What is the status of the following conjecture:
"... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one ...
0
votes
1answer
160 views
Siegel set in SO(n,1) modulo integer points?
I wonder what is known about a fundamental region for SO($n,1$) modulo its integer points? is there only one cusp? and if one writes a Siegel set in the form of
$K A_\tau N_c$, where $N_c$ is compact ...
18
votes
2answers
215 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. ...
5
votes
1answer
273 views
Hyperbolic 3-manifolds with no geometrically finite structure
Does there exist a compact hyperbolic 3-manifold $M$ that is not diffeomorphic to a geometrically finite hyperbolic manifold? If yes, can such $M$ have incompressible boundary?
I think the answer ...
4
votes
2answers
199 views
Hyperbolic Brunnian links and rectangular cusp shapes
My question is as follows.
Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components?
Here is what I mean:
The Borromean rings form a famous link $B$ (a smooth ...
3
votes
0answers
138 views
reference request: “p-adic” presentation of surfaces
On several occasions I heart about the following result:
For "certain" lattices $\Lambda$ in $SL_2(\mathbb{R})$, and almost any prime $p$ there exists a lattice $\Gamma$ in $SL_2(\mathbb{R})\times ...
3
votes
1answer
163 views
Flows in word-hyperbolic groups
I was wondering if there is a good notion of flows in word-hyperbolic groups (maybe I should say flows in the Cayley graphs of word-hyperbolic groups).
More precisely, I wonder if there is an ...
6
votes
1answer
361 views
If rational points are like entire curves, then what do algebraic points correspond to
I read somewhere that if $X$ is a projective variety of general type over a number field $K$, then rational points are an analogue of entire curves $\mathbf{C}\to X^{an}$ (with $X^{an}$ the ...
8
votes
1answer
245 views
Geodesic cuffs of pairs of pants in a hyperbolic manifold- why are they disjoint?
I'm trying to understand Kahn-Markovic's celebrated Immersing almost geodesic surfaces in a closed hyperbolic three manifold. There is something probably quite basic which I can't figure out.
We have ...
6
votes
0answers
248 views
Closed geodesics on a closed, negatively curved Riemannian manifold
I have been searching for a while for a proof of the following fact: For a closed Riemannian manifold, all of whose sectional curvatures are negative, each free homotopy class of loops contains a ...
