All Questions
Tagged with gr.group-theory hyperbolic-geometry
86 questions
3
votes
1
answer
158
views
Geometry and topology of Fuchsian character varieties
Consider the hyperbolic space, $\mathbb H^2$. A Fuchsian group is a discrete subgroup of $\text{PSL}(2,\mathbb R)$. We can generate tessellations, especially $\{p,q\} \;\text{tesellations}$ of $\...
- 357
3
votes
1
answer
177
views
Lengths of generators of surface group
Let $\Sigma$ be a closed genus $g\geq 2$ Riemann surface, which we equip with its unique constant curvature $-1$ hyperbolic metric. Let $\pi_1(\Sigma)$ be its fundamental group with respect to some ...
- 12.3k
6
votes
1
answer
146
views
If $X$ is a hyperbolic, locally finite graph with $\partial X \cong S^1$, and $G$ acts cocompactly but not properly on $X$, what can we say?
It is an important and deep fact of geometric group theory that if the Gromov boundary of a hyperbolic group $G$ is a circle, then $G$ is virtually Fuchsian [Tukia, Gabai, Casson-Jungreis...]. I am ...
- 639
11
votes
1
answer
270
views
Example of three dimensional atoroidal Poincaré duality group with some pathology
I am looking for a 3-manifold which is closed, aspherical, orientable, and atoroidal. And additionally I want to see an example that does not admit a fixed-point-free action on a simplicial tree. As a ...
- 28.2k
1
vote
1
answer
162
views
Divergence functions in hyperbolic groups
Gromov hyperbolicity has many characterizations, one of them being the existence of a super-linear divergence function, see definition below.
We note that in $\mathbb{R}^2$ there is no divergence ...
- 8,401
5
votes
1
answer
206
views
Solving equations in hyperbolic groups and subgroups of isometry of a Gromov hyperbolic space
Let $\Gamma$ be a hyperbolic group. Let $g$, $\gamma\in \Gamma$ freely generate a non-abelian semigroup (in particular, they don't commute and have infinite order). Does the equation $g\gamma^n=h^m$ ...
- 63.9k
7
votes
2
answers
278
views
Which pairs of conjugates of $\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$ generate $\operatorname{SL}(2,\mathbb{Z})$?
When do two distinct conjugates of $U := \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ generate $\DeclareMathOperator\SL{SL}\SL(2,\mathbb{Z})$? The classic example is $U,L^{-1}$, where $L = \...
- 15.4k
2
votes
1
answer
244
views
Markov property for groups?
My question again refers to the following article:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, arXiv:...
- 2,090
13
votes
0
answers
223
views
Are there free and discrete subgroups of $\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$ that are not Schottky on any factor?
$\DeclareMathOperator\SL{SL}$Does there exist a free and discrete subgroup $\Gamma < \SL_2(\mathbb{R}) \times \SL_2(\mathbb{R})$ such that neither $\pi_1(\Gamma)$ nor $\pi_2(\Gamma)$ is free and ...
- 680
7
votes
1
answer
600
views
Examples of groups that are unknown to be acylindrically hyperbolic
Let $G$ be a group. We say that $G$ is acylindrically hyperbolic (for short, AH) if $G$ admits an isometric, acylindrical, and non-elementary action on some Gromov hyperbolic space $X$.
Here is the ...
- 8,401
5
votes
1
answer
242
views
Cancellation of elements in the Gromov boundary of a free group
Let $A$ be a finite set of free generators and their inverses and $F$ the free group generated by elements in $A$ (some call $A$ the alphabet of $F$). For each $g\in F$, use $\vert\,g\,\vert$ to ...
- 307
2
votes
2
answers
464
views
Combination theorems for discrete subgroups of isometry groups
Maskit's combination theorem says: if $M=M_1\cup_\Sigma M_2$ is a union of hyperbolic 3-manifolds $M_1=\Gamma_1\backslash H^3, M_2=\Gamma_2\backslash H^3$ along a surface $\Sigma$, and if the limit ...
- 4,713
3
votes
0
answers
463
views
Representations of triangle groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up.
Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...
- 63.9k
4
votes
1
answer
212
views
Infinitely divisible elements in Gromov hyperbolic groups
An element $g\in G$ in a group $G$ is called infinitely divisible if $b=y^n$ for infinitely many different $n\in {\Bbb Z}$. It is not hard to find a finite CW-complex (or even a compact manifold) ...
- 28.2k
7
votes
1
answer
505
views
Rational stable translation length
Let $G$ be a finitely generated group and $S$ a finite generating set and consider the word metric associated to $S$.
If $g\in G$, define its stable translation length as $l(g)=\lim_n \frac{d(e,g^n)}{...
- 8,401
12
votes
2
answers
2k
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?
For ...
- 63.9k
3
votes
1
answer
375
views
Reference for triangle groups
Can anyone suggest to me some references for studying triangle groups? Especially the existence of finite index subgroups, subgroups isomorphic to fundamental groups of compact surfaces etc.
- 12.9k
1
vote
1
answer
239
views
Area of fundamental domain of Fuchsian group and index of a Fuchsian group in the triangle group
Let $\mathbb{H}$ be the upper half plane model of hyperbolic geometry. Let $\Gamma$ be the Fuchsian group such that $\mathbb{H}/\Gamma$ is the compact orientable surface of genus $2$.
Suppose $\Gamma =...
- 63.9k
30
votes
7
answers
5k
views
Why are Fuchsian groups interesting?
I keep hearing that fuchsian groups are interesting for other reasons than the Fuchsian model for hyperbolic Riemann surfaces.
What are those reasons?
Are the Fuchsian groups with fixed points ...
- 28.2k
0
votes
0
answers
164
views
Classification of fundamental domains of a fuchsian group
Let $G$ be the (2,3,7) triangle group. We can see it as symmetry group of (2,3,7) tiling of the hyperbolic plane or symmetry group of $[3^7]$ tiling of the hyperbolic plane. This contains translations,...
- 63.9k
8
votes
1
answer
487
views
When are groups generated by reflections in a triangle discrete?
Take a triangle in the (Euclidean or hyperbolic) plane, and consider the group of isometries generated by the reflections in the three sides of the triangle. If the angles between adjacent sides are ...
- 68.9k
3
votes
1
answer
458
views
Parabolic elements and hyperbolic elements in SL(2,R)
Let $\Gamma \subset \mathrm{SL}(2,\mathbb{R})$ be a lattice. If $N_1, N_2$ are a pair of independent parabolic subgroups contained in $\Gamma$, why must $\Gamma$ contain a hyperbolic element? By ...
- 28.2k
1
vote
1
answer
79
views
Let $H$ be a co-dimension 1 quasiconvex subgroup of a one-ended hyperbolic group $H$. Does $H$ permute the components of $\partial G - \Lambda H?$
Let $H$ be a co-dimension 1 quasiconvex subgroup of a one-ended hyperbolic group $G$. In particular, $\partial G$ is connected but $\partial G - \Lambda H$ is disconnected. The number of components of ...
- 8,401
8
votes
1
answer
155
views
Are the non-free factors of Grushko decomposition of a finitely generated convex-cocompact (but not cocompact) subgroup of PSL$(2,\mathbb{R})$ finite?
See Grushko decomposition theorem.
Are the non-free factors of Grushko decomposition of a finitely generated convex–cocompact (but not cocompact) subgroup of $\operatorname{PSL}(2,\mathbb{R})$ finite?
...
- 15.4k
3
votes
1
answer
165
views
Reference request for statement concerning free subgroups of $ \mathrm{SL}_2(\mathbb{Z}). $
I am interested in finding a reference for the following claim:
There exists a free subgroup $F_2$ of $\mathrm{SL}_2(\mathbb{Z})$ on two generators that does not contain any nontrivial unipotent ...
- 28.2k
2
votes
1
answer
136
views
Example of maximal multicurve complex
in this paper we have :
" On the Teichmüller tower of mapping class groups By Allen Hatcher at Ithaca, Pierre Lochak at Paris and Leila Schneps."
Definition. The maximal multicurve complex $...
- 204
1
vote
0
answers
68
views
Variation of the geometry of a Dirichlet region as the defining point varies
Let $\Gamma$ a Fuchsian group acting on the hyperbolic plane $\mathfrak{H}$. For me, I am most interested in the case where $\Gamma$ has a fundamental domain that is a finite-polygon with all ...
- 5,349
6
votes
1
answer
375
views
Non-compact Dirichlet fundamental domains and free Fuchsian groups
Let $G$ be a finitely generated Fuchsian group, and let $\mathcal{F}$ denote the Dirichlet fundamental domain of $G$ with respect to $0$ in the Poincaré disc model.
Assume throughout that $\mathcal{F}$...
- 28.2k
7
votes
1
answer
458
views
A criterion for loxodromicity in Gromov-hyperbolic spaces
Recall that an isometry of a Gromov-hyperbolic space $X$ is called loxodromic if it has exactly two fixed points on the Gromov boundary $\partial X$, one being "attracting" and the other &...
- 63.9k
1
vote
0
answers
142
views
Ideal Ford domain (for finite index subgroup)
Let $G$ be a lattice Fuchsian group with parabolic elements, seen as a discrete subgroup of matrices
$
g=
\begin{pmatrix}
\alpha & \overline{\beta} \\
\beta & \overline{\alpha}
\end{pmatrix}
$...
- 33
2
votes
1
answer
219
views
Dirichlet region of a free group
Let $G$ be a non-uniform lattice Fuchsian group and let $P$ be a Dirichlet region for $G$. In particular $G$ has parabolic elements, $P$ is not compact and has finite area. We are in the unit disc. Is ...
- 28.2k
3
votes
0
answers
80
views
Presentations of $\mathbf{PGL}_3(\mathbb{F}_q)$ by three involutions
If I am not mistaken, the group $\mathbf{PGL}_3(\mathbb{F}_q)$, with $\mathbb{F}_q$ the finite field with $q$ elements, can be generated by three involutions.
Where could I find such representations ?
...
- 4,547
1
vote
1
answer
101
views
Presentations of $\mathbf{𝐏𝐆𝐋}_3(\mathbb{F}_2)$ by three involutions, 2
I am searching for a presentation of the group $\mathbf{PGL}_3(\mathbb{F}_2)$ for which the generators are involutions $a, b, c$, and such that the following relations are present [among extra ...
- 44.4k
8
votes
1
answer
572
views
Mostow Rigidity Theorem and reconstruction from fundamental group
The Mostow Rigidity Theorem is phrased in terms of a relationship between isometries and isomorphisms of fundamental groups, which raises an obvious question. Given the fundamental group of a complete ...
- 591
4
votes
2
answers
486
views
When existence of loxodromic, WPD elements implies an action is acylindrical
Definitions
Say that $(X,d)$ is a $\delta$-hyperbolic space and that $G$ is a finitely generated group acting on $X$ by isometries.
Recall that an action of $G$ on $X$ is called acylindrical if the ...
- 8,401
5
votes
1
answer
446
views
The stabilizers of the canonical boundary action of hyperbolic groups
My question is that
Is every stabilizer of the canonical boundary action of a hyperbolic group on its Gromov boundary a finitely generated group?
I guess every stabilizer is a (finitely generated) ...
- 2,090
7
votes
1
answer
372
views
Thickness and hierarchical hyperbolicity
Thick metric spaces were introduced by Behrstock, Drutu and Mosher, see here. Hierarchically hyperbolic spaces were introduced by Behrstock, Hagen and Sisto, see here.
I've heard that it is open ...
- 312
4
votes
1
answer
128
views
Copies of $\mathbb{Z}\oplus \mathbb{F}_2$ in non-affine, irreducible Coxeter groups
Let $\left(W,S\right)$ be a non-affine, irreducible Coxeter system and assume that $W$ contains a copy of $\mathbb{Z}\oplus\mathbb{Z}$ (this is equivalent to $W$ being not word hyperbolic). Does this ...
- 1,016
2
votes
1
answer
198
views
Equivalence of harmonic measures on hyperbolic groups
Consider a Gromov-hyperbolic group $\Gamma$ and let $\mu$ be a finitely supported probability measure on $\Gamma$. Assume that the support of $\mu$ generates $\Gamma$ as a semi-group, in other words, ...
- 2,090
5
votes
1
answer
342
views
Relations between boundaries of groups acting on hyperbolic spaces with WPD elements
Let $(X,d)$ be Gromov-hyperbolic space and let $\Gamma$ be a finitely generated group acting on $\Gamma$ by isometries. Recall the following two definitions.
Say that the action is acylindrical if ...
- 121
6
votes
1
answer
189
views
Starting letters of equivalent infinite geodesic paths of hyperbolic Coxeter groups
Let $\left(W\text{, }S\right)$ be a Gromov hyperbolic Coxeter system and denote by $\partial W$ the corresponding Gromov boundary. For $z\in\partial W$ let $\alpha$, $\beta$ be infinite geodesic paths ...
- 8,493
9
votes
1
answer
542
views
How much do the classes of geodesic metrics and Green metrics on a Gromov hyperbolic group differ?
Background
Let $G$ be a Gromov hyperbolic group. If $G$ acts properly discontinuously and cocompactly on a proper geodesic metric space $X$, then any bijective identification of $G$ with its orbit ...
- 2,090
5
votes
0
answers
183
views
Finitely generated nilpotent groups as cusp groups
I recently learned about the following question, asked by I. Kapovich :
Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\geq 3$...
- 2,090
9
votes
2
answers
886
views
Hyperbolic $3$-manifold groups that embed in compact Lie groups
Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group?
It is known that every surface group can be embedded into any semisimple ...
- 68.9k
6
votes
1
answer
564
views
What are the $2 \times 2$ matrix generators of $\text{SL}_2\big(\mathbb{Z}[i]\big)(2+i)$?
I have been trying to learn about congruence groups. Here is an example:
\begin{eqnarray*} \Gamma\big(1+2i\big) &=& \text{SL}_2\big(\mathbb{Z}[i]\big)(1+2i) \\ \\
&=& \left\{ \left(...
- 5,259
10
votes
1
answer
738
views
Parabolic subgroups of relatively hyperbolic and CAT(0) groups
Let $G$ be a finitely generated group. We say that $G$ is CAT(0) if it acts properly and co-compactly by isometries on a CAT(0) space.
We say it is hyperbolic relative to a collection $\Omega$ of ...
- 8,401
-1
votes
1
answer
160
views
How to understand this isomorphism? [closed]
A Proposition from a book written by Benson Farb and Dan Margalit
- 96.4k
17
votes
5
answers
4k
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 ...
- 28.2k
3
votes
2
answers
159
views
What is the homeomorphism from $\Gamma \backslash T_1 \mathbb{H}$ to $T_1(\Gamma \backslash \mathbb{H})$
Let $\mathbb{H}$ be hyperbolic plane, $\Gamma$ is a discrete subgroup of $PSL_2(\mathbb{R}$) so that $\Gamma \backslash \mathbb{H}$ is a compact hyperbolic surface. Maybe it will be very simple to you ...
15
votes
2
answers
2k
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 ...
- 1,713