1
vote
1answer
21 views

Question about particular words in the free group on three generators

In the free group generated by the letters $x,y,z$ suppose that we have a word such that for any one of $x,y,z$ the indices of each occurrence of that letter in our word sum to zero. Suppose further ...
2
votes
2answers
29 views

Cocompact group actions have cobounded orbits

Assume $X$ is a complete, locally compact, geodesic metric space (in particular, $X$ has closed balls are compact, by the Hopf-Rinow Theorem). Assume $G$ acts isometrically on $X$. We say $Q\subset X$ ...
1
vote
0answers
37 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 ...
3
votes
1answer
94 views

Any subgroup of an abelian group is undistorted.

I need some help with the following math problem. I am studying some notes on Geometric Group Theory and I came across the following problem. Prove: Any subgroup of an abelian group is ...
2
votes
0answers
33 views

If $H$ is a subgroup of $G$, then the inclusion is a quasi-isometry if and only if $H$ has finite index in $G$.

If $H$ is a subgroup of $G$, then the inclusion is a quasi-isometry if and only if $H$ has finite index in $G$. I am reading through some notes on Geometric Group Theory and I came across this side ...
2
votes
1answer
27 views

What does it mean for a group to act cocompactly by isometries on a topological space $X$?

What does it mean for a group to act cocompactly by isometries on a topological space $X$? I know if $X$ is a topological group, and $A$ a subspace, then $A$ is cocompact iff $X/A$ is compact. Not ...
2
votes
2answers
40 views

Is a $0$-hyperbolic group free?

In his article, Abderezak Ould Houcine asks the following question: If $G$ is a hyperbolic group, let $\delta_0(G)$ denote the infinimum of $\delta$ for which $G$ is $\delta$-hyperbolic. When ...
4
votes
1answer
78 views

Free product as automorphism group of graph

Let $A$ and $B$ be two groups. We define following graph $X$. The set of vertices is the left cosets $gA$ and $gB$ where $g\in A*B$ (By $A*B$, I mean the free product of $A$ and $B$). The edges of the ...
1
vote
1answer
86 views

Trivial elements in $T(a,b,c)$

Consider the group $T(a,b,2)=<x,y|x^a, y^b, (xy)^2>$ and assume none of $a$ or $b$ is equal to $2$. How can one list all the trivial words (say up to length $11$ and apart from $(xy)^{2n})$) in ...
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
26 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
30 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
232 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. ...
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 ...
0
votes
1answer
35 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
33 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
61 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
110 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 ...
3
votes
1answer
68 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 ...
2
votes
2answers
100 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
103 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? ...
2
votes
2answers
81 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 ...
3
votes
2answers
64 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
98 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?
13
votes
1answer
298 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
415 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 ...
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 ...
0
votes
1answer
52 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 ...
17
votes
1answer
299 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 ...
2
votes
2answers
198 views

HNN extensions as fundamental groups

I have heard that the Seifert–van Kampen theorem allows us to view HNN extensions as fundamental groups of suitably constructed spaces. I can understand the analogous statement for amalgamated free ...
1
vote
1answer
100 views

Outer automorphism group of a free product

Suppose $G=\mathbb{Z}\ast C_n=\langle a, b; b^n\rangle$ is the free product of the infinite cyclic group with a finite cyclic group. Then $G$ has finite outer automorphism group. However, my proofs ...
2
votes
0answers
77 views

The program “gm” by Epstein and Rumsby for drawing tesselations and Cayley graphs

In the book Word Processing in Groups by David Epstein, there is a pair of pages, 38 and 39, which have two pictures on them. If you are familiar with the book, you probably know exactly what I am ...
3
votes
1answer
113 views

normalizer of a cyclic subgroup in a torsion-free hyperbolic group

How one can show that in a torsion-free hyperbolic group if elements $x$ and $y$ (edit: $y\ne1$) satisfy: $$ xy^mx^{-1}=y^n $$ then $m=n$ and $x$ and $y$ belong to the same cyclic subgroup?
2
votes
1answer
56 views

Embedding and Graph(Tree) of Groups

According to Serre's definition (in Serre's Trees): ($G$,$T$) is a tree of groups if T is a tree and there are groups $G_v$ and $G_e = G_\bar e $ for each $v\in vertT$ and $e \in edge T$, where ...
5
votes
0answers
149 views

Higman group with 3 generators is trivial

The group $G$ generated by $x,y,z$ subject to the relations $[x,y]=y$, $[y,z]=z$, $[z,x]=x$ is trivial. This isn't the case for the corresponding group with 4 generators, which is the famous Higman ...
5
votes
0answers
200 views

Combinatorial total space for finitely generated torsion-free groups?

Motivation: I'm an operator algebraist and I'm looking for an answer to the main question in order to build non-trivial spectral triples for a class (as large as possible) of discrete groups. $\to$ ...
4
votes
3answers
178 views

metric property in a group

Can we define a metric function on a group $G$? Please give examples other than $\mathbb{R}$. Actually most groups have elements in discrete manner. It sounds vague but I can't be more precise.
2
votes
0answers
103 views

Subgroups of amalgamated free product

My question is the following: Suppose that we are given the amalgamated product $ G = G_1 * _{G_3} G_2 $, and subgroups $ H_i \le G_i $ for $i=1,2,3$, such that in addition $H_3$ is as large as ...
2
votes
1answer
64 views

Double cover of symplectic groups

What is the normal definition of double cover of Symplectic group? I couldn't find a simple and understandable definition
2
votes
1answer
159 views

Subgroup of a virtually cyclic group

Let $G$ be a virtually cyclic group, i.e., G has an infinite cyclic subgroup $H$ of finite index. Is it true that if $H'$ is another infinite cyclic subgroup of $G$ then $H'$ must be of finite index ...
1
vote
1answer
67 views

Finitely generated group and quadratic isoperimetric functions.

It is well known that a finitely generated group with simply connected cones has isoperimetric functions. Papasoglu in the paper "On the Asymptotic cone of groups satisfying a quadratic ...
8
votes
0answers
131 views

How useful are geometric aspects when studying finite groups?

My newbie impression when studying finite group theory is that geometric aspects are not very prevalent. Cayley graphs play quite some role for visualising finite groups, but compared to the study of ...
2
votes
2answers
96 views

Question about representation of free products of groups.

Does anyone have an idea of books or papers that treats representation theory of free products of groups? What properties of factors of of a free product suggest a possible representation? ...
4
votes
2answers
226 views

Quotient of Cayley graph of the free group on two generators by a subgroup.

If $F=F(\{a,b\})$ is the free group on two generators $a$ and $b$ and $G$ is the subgroup $$G=\:\langle b^n a b^{-n}|\: n\in \mathbb{N}\rangle \leq F$$ I am trying to work out what the quotient graph ...
2
votes
1answer
101 views

Proving that commensurability is transitive

We have that two groups $\Gamma$ and $\Gamma'$ are commensurable if there exist finite index subgroups $G \leq \Gamma$ and $G' \leq \Gamma'$ such that $G \cong G'$. We denote this $\Gamma \approx ...
2
votes
0answers
51 views

A problem about normalizers in $PSL(V)$

Let $K=\mathbb F_{p^k}$ a finite field, and $V$ a vector space on $K$. Clearly $PSL(V)=SL(V)/SL(V)\cap Z$ acts on $V$ by the following rule ($Z$ is the subgroup of the scalar functions): ...
17
votes
3answers
466 views

Prove an inequality on $l^2$ sequences over $F_2$

Denote $F_2$ the free non-abelian group on two letters $a, b$. Note that any element in $F_2$ is just a word formed by letters from the set $\{a,b,a^{-1},b^{-1}\}$, and the group structure is given ...
13
votes
2answers
362 views

H0w have group theory and fractal geometry been combined?

Has there been a significant tie made between group theory and fractal geometry? What are some ways that they have been tied together? I've been inspired to ask this question by this image of a free ...
2
votes
1answer
116 views

Systems of coset representatives

I have the following question. Let $H\leq U\leq G$ be (not necessary finite) groups. Let $S$ is a System of Right coset represantatives of $U$ in $G$, i.e. $\bigcup_{s\in R} Us=G$ with $Us\cap ...
3
votes
0answers
65 views

Classification of Bieberbach groups

Does anybody know if there exists a list of the four dimensional Bieberbach groups presented by generators and relations on the web?. I know there exists the book Crystallographic Groups of ...