6
votes
0answers
96 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
1answer
227 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
1answer
112 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
1answer
97 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
1answer
112 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
1answer
117 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
1answer
262 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?
8
votes
1answer
312 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
1answer
345 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
53 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 ...
7
votes
1answer
320 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
1answer
455 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
1answer
216 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
2answers
237 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
0answers
140 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
0answers
107 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 ...
2
votes
2answers
220 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
1answer
307 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
2answers
633 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
3answers
564 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
2answers
547 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
1answer
654 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
2answers
276 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
1answer
354 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
2answers
873 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
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 ...