# Tagged Questions

**6**

votes

**0**answers

108 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 ...

**4**

votes

**1**answer

243 views

### What is known about maximal free subgroups of surface groups?

Let $\Gamma_g=< a_1,...,a_g,b_1,...,b_g | \prod_{i=1}^g [a_i,b_i]>$ (a surface group). What is known about maximal free subgroups of $\Gamma_g$ for $g>1.$ (I.e. free subgroups which are not ...

**2**

votes

**1**answer

116 views

### Density of ends of long words in a hyperbolic group

Let $G$ be a Gromov hyperbolic group with a generating set $S$. For each $g \in G$, let $\xi_g$ be the point in the ideal boundary corresponding to the sequence $(g^n)$. Let $l_S(g)$ be the word ...

**3**

votes

**1**answer

102 views

### Fixed directions and Zariski density of hyperbolic groups

It is a fact that if $\Lambda$ is a nonelementary subgroup of ${\rm PSL_2}(\mathbb{C})$ which contains an hyperbolic transformation and moreover ${\rm tr}(g)\in\mathbb{R}/\pm 1$ for all $g\in\Lambda$ ...

**3**

votes

**1**answer

120 views

### Classification of maximal nonuniform Fuchsian lattices existent?

I am interested in the set of all non-cocompact Fuchsian lattices which all have a distinguished point as cusp, say $\infty$ in the upper half plane model of the hyperbolic plane. Of course, the ...

**3**

votes

**1**answer

137 views

### Arithmetic Fuchsian lattices that are not finite index subgroups of Eichler orders?

Lindenstrauss' proof of AQUE (arithmetic quantum unique ergodicity) assumes that the Fuchsian lattice is an Eichler order or, if I understand it correctly, a finite index subgroup of an Eichler order. ...

**5**

votes

**1**answer

270 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?

**9**

votes

**1**answer

385 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 ...

**12**

votes

**1**answer

377 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

**0**answers

56 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 ...

**8**

votes

**1**answer

348 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 ...

**9**

votes

**1**answer

469 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 ...

**1**

vote

**1**answer

218 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 ...

**4**

votes

**2**answers

265 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 ...

**3**

votes

**0**answers

141 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 ...

**2**

votes

**0**answers

109 views

### Local curvature in a Cayley complex

I'm curious about non-shperical regions in the Cayley complex of a hyperbolic group $G = < X : R \>$ that one might reasonably consider "curved", but I haven't found much discussion of local ...

**1**

vote

**2**answers

234 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**

votes

**1**answer

320 views

### Does the fundamental group of a surface have rigid subgroups?

Suppose $\Gamma$ is the fundamental group of a closed, oriented surface $S$. Let $B$ be a finitely generated, infinite index subgroup of $\Gamma$, and let $\Gamma_B$ be the compact core of the ...

**16**

votes

**2**answers

659 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 ...

**9**

votes

**3**answers

593 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 ...

**3**

votes

**2**answers

557 views

### A construction of generators of discrete subgroups of SL(2,R)

I know about geometrical method of construction of discrete subgroups of $SL(2,\mathbb{R})$ using Lobachevsky plane (e.g. B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry --- Methods and ...

**12**

votes

**1**answer

683 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 ...

**5**

votes

**2**answers

277 views

### Subgroup structure of $\mathrm{SO}(1,n)_0$

A classification (up to conjugacy) of all closed subgroups of the (identity component of the) Lorentz group $\mathrm{SO}(1,3)_0$ in terms of the subalgebras of its Lie algebra was given in
R. Shaw. ...

**12**

votes

**1**answer

362 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 ...

**9**

votes

**2**answers

925 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?
...

**18**

votes

**5**answers

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 ...