1
vote
0answers
38 views

Let Cay(G, S) be the cayley graph of G with respect to the finite generating set S where G=⟨S∣R⟩ and R is finite.

Let $\operatorname{Cay}(G, S)$ be the cayley graph of $G$ with respect to the finite generating set $S$ where $G = \langle S\mid R\rangle$ and $R$ is finite. I am reading some notes that claim that ...
0
votes
1answer
39 views

Limit set of Kleinian group

Let $\Gamma \subset PSL_2 (\mathbb{C})$ a Kleinian group coming from a discrete faithful representation $\rho : \pi_1(M) \to PSL_2 (\mathbb{C})$ of the fundamental group of a closed connected ...
3
votes
1answer
53 views

Constructing The Cayley Graph and quasi-isometry to $\mathbb{Z}$

If we have a group $G$ defined by: $G=\langle a,b\mid b^2=1\rangle$ then I first need to construct the cayley graph of this, now I think that this is going to look like the "telephone pole" metric ...
0
votes
0answers
48 views

Showing that triangles in $\mathbb{Z}$ are thin

If we let $\mathbb{Z}$ be generated by $\{3,5\}$ then I have a question which asks me to show that geodesic triangles are $k$-thin and to find a minimum bound on $k$. I have been thinking about this ...
1
vote
1answer
30 views

What is PSU(1,1) PSU(2,1)? Is there a general group of the form PSU(n,m)?

I have seen it written about groups PSU(1,1), PSU(2,1). But what exactly are these? The definitions were not given, and I can't seem to find a definition online. Moreover is there a general class of ...
3
votes
2answers
82 views

Shortest words in a group with finite presentation

Suppose we're given a group with presentation G=, where both the generating set and the relations are finite. Given a word $w$ in the elements of $X$, I would like to know whether this word is ...
3
votes
1answer
128 views

Fuchsian groups and topological isomorphism

I have a (finite) presentation of a group and I am wanting to prove that it is not Fuchsian. Because it is given by a presentation, a neat, algebraic description of Fuschian groups would be nice. This ...
0
votes
0answers
45 views

boundary of word-hyperbolic group

Let $G$ be a word-hyperbolic group and let $\partial G$ be its (Gromov) boundary. Do there exist criteria that imply that all non-trivial finite order elements of $G$ act fixed-point freely on ...
2
votes
1answer
81 views

Locally cyclic subgroups of a hyperbolic group

How can we show that locally cyclic subgroups (ie. groups whose finitely generated proper subgroups are cyclic) of a hyperbolic group are cyclic?
3
votes
0answers
56 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 ...
0
votes
1answer
96 views

Question on proof in “Primer on MCGs”

This is a question about the proof of Proposition 1.4 in Farb and Margalit's "Primer on Mapping Class Groups" (in v. 5.0, it is on page 37 in the PDF, which you can download here). The proposition ...