Geometric group theory is the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act. Consider using with the (group-theory) tag.

learn more… | top users | synonyms

2
votes
1answer
49 views

Representation of cell location in hyperbolic plane

I want to represent an order-5 square tiling (image from Wikipedia; more text below image): Obviously for a simple grid I can uniquely refer to a given square by its ...
0
votes
0answers
42 views

free subgroup of amalgamated free product groups

Suppose $G=A*_{C}B$ be an amalgamated free product group, in general, I am interested in the question whether $G$ contains a free subgroup, and given specific examples, I always first calculate its ...
5
votes
1answer
50 views

General introduction to orbifolds?

Where should I go to learn about orbifolds? I am interested in a general introduction that gives precise definitions and clear explanations. I have a fair background in topological and smooth ...
2
votes
1answer
33 views

Fundamental domain of a Fuchsian group which is not locally finite

I am trying to understand Example 9.2.5 in Beardon's book The Geometry of Discrete Groups. The goal is to construct a fundamental domain of a Fuchsian group which is not locally finite. Definitions ...
1
vote
1answer
25 views

Growth of direct products

Let $G$ and $H$ be finitely generated groups. Is it true that $ \gamma_{G \times H} $ is weakly equivalent to $ \gamma_{G}\gamma_{H} $? Assuming the answer is yes, can one use exactly the same ...
1
vote
1answer
23 views

Are free products of exponential growth?

Let $G$ and $H$ be finitely generated groups. Is the free product $G*H$ always of exponential growth? I am guessing that the answer is yes, but I don't know how to prove it correctly. My idea is to ...
1
vote
1answer
27 views

Question about Lamination for free groups

I have a question about the leaves of "Stable lamination" as defined in the paper "LAMINATIONS, TREES, AND IRREDUCIBLE AUTOMORPHISMS OF FREE GROUPS" and the periodic conjugacy classes. More ...
5
votes
2answers
227 views

Showing that a group with a presentation is free/not free

Show that the group with presentation $\langle a, b, c \mid a^2cb^3\rangle$ is free with basis $\{a, b \}$. Show that the group with presentation $\langle a, b, c \mid a^3b^3 \rangle$ is not free. ...
1
vote
0answers
50 views

Showing that $\mathbb{R}\times [0,1]$ is quasi-isometric to $\mathbb{R}$

I have a question that asks me to show that $\mathbb{R}\times [0,1]$ is quasi-isometric to $\mathbb{R}$ I have having trouble showing what I have is a quasi isometry. My map is simply: ...
3
votes
0answers
38 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
45 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
40 views

$T_3$ is quasi isometric to $T_4$

I have a question which asks me to show that $T_3$ is quasi isometric to $T_4$, that is the three and 4 valence trees. I know that this means that I have to define a map $f:T_3\rightarrow T_4$ such ...
0
votes
1answer
33 views

Cayley graph of an amalgam

I was wondering what the Cayley graph of an amalgam $A*B$ (over $\{ 1\}$ ) is? $A, B$ are finitely generated. Can't work it out. Thanks!
0
votes
0answers
31 views

Dehn presentation question

I have just shown that if a group $G$ admits a Dehn presentation then there are finitely many conjugacy classes of finite order. I'm then trying to deduce from that fact that there is some $N$ such ...
6
votes
1answer
58 views

Baumslag–Solitar $B(1,2)$ is not hyperbolic

I have a question which asks me to show that the Baumslag–Solitar $B(1,2)$ is not hyperbolic by considering its Cayley graph and showing that triangles can be arbitrarily fat. The Cayley graph can be ...
6
votes
0answers
104 views

Applications of Cayley Graphs in Physics

I have been recently reading about Cayley graphs and character theory. It is evident that Cayley graphs are very useful tool in theoretical computer science. In physics, Cayley graphs seem do appear ...
2
votes
1answer
51 views

Quasi-isometry of Schreier coset graphs

Let $G$ be a finitely generated group and $H$ a subgroup. For any choice of a generating set $X$ we can form the Schreier coset graph. Is this graph independent of the generating set up to ...
3
votes
1answer
67 views

3-Dimensional analogue of a Dihedral Group

I'm reading about dihedral groups right now, and I'm being asked to find the orders of various groups of rigid motions of 3-dimensional polygons. What I want to know, though, is if there's a nice ...
3
votes
2answers
74 views

Is the fundamental group of a compact manifold finitely presented?

Let $X$ be a connected compact smooth manifold. If $X$ is boundaryless, we can choose a Riemannian metric for $X$ so that $\pi_1(X)$ acts geometrically (ie. properly, cocompactly, isometries) on the ...
0
votes
1answer
35 views

How to find linear fractional transformations of a group

Finite presentation of G is $$\langle x,y,t,q : x^2=y^3=t^2=q^2 =1,tq=qt,ty=yt,qyq=y^{−1},xt =qx \rangle.$$ I am interested in finding linear fractional transformations x,y,t,q which satisfy all its ...
1
vote
0answers
55 views

Does there exists a finitely presented group with Dehn function > n^3 and all asymptotic cones simply connected

it is well known that all asymptotic cones simply connected implies polynomial Dehn function (Gromov). It is also well known that quadratic Dehn function implies all asymptotic cones simply connected ...
4
votes
2answers
132 views

Intuition behind the failure of unimodularity

If $G$ is a locally compact group then up to normalization it admits a unique Haar measure: a left invariant measure defined on all Borel subsets of $G$, which assigns every compact set a finite ...
2
votes
2answers
93 views

Fundamental group of a Cayley graph

Could someone please indicate the proof of the following fact (I believe strongly, but I am not sure, that this is true). Let $F$ be a free group of finite rank, $N<F$ a normal subgroup and ...
2
votes
1answer
100 views

Is there a way to show that two groups are isomorphic by visual representation(Cayley diagram)?

I got a question asking me to prove that $V_4$ and $C_2 \times C_2$ are isomorphic. I can do this algebraically. However, I am curious if there is there a way to explain this using the diagram? ...
1
vote
2answers
44 views

Immersions when the target space isn't a differentiable manifold (but *almost* is)

I've come across this situation in a number of places but it's most glaring in the lecture notes I'm currently reading. (PCMI lectures on the geometry of outer space). We have a map from a circle ...
2
votes
2answers
75 views

Actions of Finite Groups on Trees

Any action of a finite group on a (non-empty) tree has a global fixed point (in the sense that there is a vertex fixed by all group elements or an edge fixed by all group elements). There is a ...
2
votes
1answer
41 views

Using the Gromov product in inappropriate ways

The Gromov product $(x,y)_z=1/2(d(z,x)+d(z,y)-d(y,x)$ is used in Gromov hyperbolic groups to measure how long two rays stay together or how thin a triangle is. In particular, if $(x,y)_z=n$ in a ...
3
votes
2answers
63 views

Amalgamated product of groups

This is just a problem solving question. Let $G$ be a finitely generated group such that $G=A*_CB$ where $|A:C|=|B:C|=2$ and $A,B$ are finite. Show that $G$ has a finite index subgroup isomorphic to ...
1
vote
3answers
92 views

An example of a residually finite group which is not Hopf

trying to think of any residually finite group which is not Hopf. Any help?
11
votes
1answer
285 views

On which structures does the free group 'naturally' act?

One of the best ways to get a handle on a group is to recognize it as isomorphic to a set of symmetries of some structure. The dihedral group of order $2n$ is easily recognized as the set of ...
4
votes
4answers
412 views

Presentation of a non-trivial group

I'm having a bit of trouble understanding group presentations. For example, I'm reliably informed that the group $$ \langle x, y \mid x^2=y^3 \rangle $$ is not the trivial group, but I don't see ...
0
votes
0answers
55 views

Quotient group and graphs

what is the Quotient group and how we can compute it for Petersen graph? what properties of graphs are incurred in the quotient groups of graphs? for example suppose G=(V,E) , D is the free abelian ...
2
votes
1answer
85 views

Show the existence of a certain subgroup of F2

This is a homework question: I need to find three subgroups of the free group with two generators, $F_2$, with certain properties. I have found the other two by constructing covering spaces of $S^1 ...
2
votes
1answer
50 views

find a special free subsemigroup

It is well-known theorem that for an elementary amenable group $G$, $G$ has exponential growth rate iff $G$ contains non cyclic free semigroup. Now I am interested in the following questions: Let ...
1
vote
1answer
56 views

Torsion Subgroup of Mapping class group.

What is the cardinality of finite order elements in Mapping class group of a surface $S_{g,n}$ of genus g and n boundary components. 1) If it is infinite then how can I generate a collection of ...
0
votes
1answer
50 views

A combinatorial action of a discrete group is proper if and only if it has finite vertex stabilizers

First, let me fix some definitions. The action of a group $G$ on a topological space $X$ is proper if for every compact subspace $K \subseteq X$ the set $\{g \in G \ | \ g K \cap K \neq\varnothing ...
2
votes
2answers
86 views

Quasi-isometric embedding and Quasi-isometry

Let $X$ and $Y$ be geodesic metric spaces. Suppose there are quasi-isometric embeddings $f:X \rightarrow Y$, $g:Y \rightarrow X$. Then, can we say there is a quasi-isometry from $X$ to $Y$? I tried to ...
2
votes
0answers
34 views

#B(e,n) of $\mathbb{Z}^k$

notation:: #A is the number of factors of A, B(e,n)={x $\in$A|d(e,x)≦n}, and S(e,n)={x $\in$A|d(e,x)=n} Then, I want to know that #B(e,n) of $\mathbb{Z}^k$. Where $\mathbb{Z}^k$ is equipped with the ...
0
votes
2answers
114 views

The question of quasi-isometry

I cannot show the following quetion. "Let X and Y be metric spaces , and f: X → Y be a quasi-isometry.Show that there exists a finite metric space Z and a map F: X × Z → Y which is also a ...
16
votes
1answer
292 views

The kernel of free group map to surface group

$G$ is a surface group of genus $g\geq 2$ (the fundamental group of closed orientable surface of genus g). $F$ is a free group of rank $2g$ with basis $\{x_1,\dots,x_{2g}\}$. $\phi$ is a surjective ...
0
votes
1answer
70 views

Wreath product $\mathbb{Z} \wr \mathbb{Z}$ has infinite asymptotic dimension

I want to know this question that wreath product $\mathbb{Z} \wr \mathbb{Z}$ has infinite asymptotic dimension. In my text , $\mathbb{Z} \wr \mathbb{Z}$ contains a subgroup isomorphic to ...
2
votes
2answers
83 views

Free subgroups of piecewise linear homeomorphisms of the circle

For convenience, I recall the definition of piecewise linear homeomorphism: A homeomorphism $f$ of the real line $\mathbb{R}$ is called piecewise linear if there is an increasing sequence of real ...
4
votes
0answers
151 views

Is there an analogue of outer Space to study outer automorphisms of free pro-$p$ groups?

I would like to know if there is an analogue of Culler & Vogtmann's outer space to study outer automorphisms of free pro-$p$ groups. Perhaps an initial guess of such a space would be a moduli ...
0
votes
1answer
89 views

Wreath product of finitely generated groups is finitely generated

Let $G$ and $H$ be two groups generated by finite sets $\Sigma_G$ and $\Sigma_H$, and let $W=G \wr H$ be the wreath product of $G$ and $H$. Show that $W$ is finitely generated by $\Sigma_G \times ...
2
votes
1answer
103 views

The image of homomorphism of fundamental group of closed surface

$\phi: \pi_1(S)\to \pi_1(S)$ is a homomorphism of fundamental group of closed orientable surface $S$ of genus $\geq 2$. If $\phi$ is not an epimorphism, can we find a non-surjective self map $f: S\to ...
2
votes
0answers
37 views

Reference request for an explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface

Let $\Sigma_g$ be a geuns $g$ Riemann surface with $g \geq 2$. It can be thought of in the following way: it is the quotient space $$\mathbb{H}/\pi_1(\Sigma_g)$$ where an element of ...
4
votes
2answers
134 views

free subgroups of $SL(2,\mathbb{R})$

In the example section of the wikipedia article on the the Ping Pong lemma, you can see how to construct a free subgroup of $SL(2,\mathbb{R})$ with two generators $$ a_1 = \begin{pmatrix} 1 & 2 ...
0
votes
1answer
47 views

Splitting of a surface group over the subgroup associated to a closed geodesic

Let $G=\pi_1 (S)$ for a closed surface $S$, consider a closed geodesic $c$ on $S$ and let $H$ be the subgroup of $G$ induced by $c$. Is it true that $G$ splits over $H$, i.e. $G$ can be written as a ...
0
votes
0answers
58 views

Infinite geodesic rays leaving a K-quasiconvex subgroup stay K-close to it.

I am going through some basic properties of $\delta$-hyperbolic spaces and groups and I am having some difficulties proving precisey some things that are anyway intuitively clear to me. Let $G$ be a ...
1
vote
1answer
177 views

Cayley graphs of Abelian groups quasi-isometrically embeddable in R^d

Are all Cayley graphs of ${\mathbb Z}^d$ quasi-isometrically embeddable in ${\mathbb R}^d$? Or, else, do they all have the same growth exponent? Is it the same true for all finitely-generated Abelian ...