3
votes
1answer
52 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
70 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
92 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
58 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
51 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
61 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
262 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
382 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
47 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
47 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 ...
15
votes
1answer
280 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
137 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 ...
0
votes
1answer
86 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
74 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
91 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
55 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
127 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
181 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
148 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
86 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
63 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
144 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
64 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
124 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
81 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
189 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
76 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
47 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
459 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 ...
12
votes
3answers
308 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
94 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
63 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 ...
15
votes
1answer
318 views

What are the symmetries of a colored rubiks cube?

Technically the symmetry group of the rubiks cube is the symmetry group of the cube with all its label peeled off. The normal rubiks cube with all its faces painted different colors has trivial ...
2
votes
0answers
32 views

maximal independent sets and maximal independent generating sets of PSL(2,p)

I'm searching for an example of a group PSL(2,p) with a "maximal independent" set of length greater than the length of a "maximal independent generating" set. All authors refer to Whiston's papers ...
5
votes
1answer
104 views

Action of a subgroup of finite index on a tree induced by an action of a group on a tree

Let $G$ be a group wich acts on a tree $\Gamma$. Then $U$ acts on $\Gamma$ for every $U\leq G$. Question: Why does the following hold? If $|G:U|<\infty$. Then the minimal $U$-invariant subtree ...
2
votes
1answer
76 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?
2
votes
0answers
61 views

Groups acting on (regular) trees with finite quotient

Let $T$ be a regular tree, and suppose that $G \leq \mathrm{Aut}(T)$ has finite quotient graph, $T / G$. Is it true (in general) that $G$ will have trivial centralizer in the full automorphism group? ...
2
votes
0answers
33 views

Generators of SL_n of a local ring of integers.

This is a follow-up to my previous question: Generators of $GL_n(\Bbb Z)$ and $GL_n(\Bbb Z_p)$ Let $\mathcal{O}_K$ be the ring of integers in a characteristic zero non-archimedean local field $K$. ...
4
votes
1answer
229 views

The Bass-Serre tree for BS(1,2)

I'm trying to find the Bass-Serre tree corresponding to the Baumslag-Solitar group $BS(1,2)$. I could find only one reference from the Internet, which I don't understand, and although I have the ...
13
votes
1answer
626 views

What is the intution behind the ping-pong lemma?

Is the ping-pong lemma a difficult characterization of free groups? Or is it just me? Does someone have a nice intuition about its idea or should I carry on staring at the statement?
4
votes
1answer
179 views

Groups quasi-isometric to $\mathbb{Z}^n$

I am interested by the following result: A groups quasi-isometric to $\mathbb{Z}^n$ is virtually $\mathbb{Z}^n$. I know the article Harmonic analysis, cohomology, and the large-scale geometry of ...
2
votes
2answers
241 views

How many small cancellation groups are there?

It is known that there are uncountably many groups with two generators. But what about the restriction to small cancellation groups? Are there countably or uncountably many small cancellation groups? ...
10
votes
2answers
927 views

Normal subgroups of free groups: finitely generated $\implies$ finite index.

I am looking at what should be a simple exercise in geometric group theory. I have reduced the problem to just completing an exercise from Hatcher, Section 1.B page 87: 7. If $F$ is a finitely ...
3
votes
1answer
61 views

Nontrivial Splitting of a subgroup $H$ of a free product $G=A*B$

Let $G=A*B$. And let $N\unlhd A$ be a normal subgroup of A. Let $H\leq G$ be the kernel of the following map: $$\Psi:A*B\to A/N*1.$$ With Kurosh's Theorem there exists a splitting $H=(H\cap A)*(H\cap ...
2
votes
1answer
145 views

Baumslag-Solitar Group $G=\langle a,t \mid tat^{-1}=a^k\rangle\cong\mathbb{Z}[1/k]\rtimes\langle t\rangle$?

Let $G$ be the Baumslag-Solitar group $\langle a,t \mid tat^{-1}=a^k\rangle$ and $$\mathbb{Z}[1/k]:=\left\{\frac{x}{k^n}\mid x\in\mathbb{Z},n\in\mathbb{N}\cup\{0\}\right\}.$$ I'm searching for an ...
4
votes
3answers
151 views

When does the Commensurator of a subgroup of a group $G$ not equal $G$?

Let $H\leq G$ be two groups. I'm interested in the Commensurator $$\mathrm{comm}_G(H)=\{g\in G : gHg^{-1} \cap H \text{ has finite index in both}\}.$$ Obviously, $\mathrm{comm}_G(H)\leq G$. I read on ...
2
votes
0answers
58 views

JSJ-decompositions of groups and 3-manifolds: a reference request

I am, for whatever reason, interested in learning about the JSJ-decomposition of groups. Having asked around a bit, it was suggested I first learn about what is happening in the manifolds and then ...
3
votes
4answers
135 views

$U\leq G$ infinite subgroup, $H\leq G$ subgroup of finite index in $G$ $\Rightarrow$ $U\cap H\neq 1$

I have the following question: If I have an infinite subgroup $U$ of a group $G$ and a subgroup $H$ of finite index in $G$ then how can I show that the intersection of $U\cap H$ is non-trivial??? ...
3
votes
1answer
212 views

Quasi-Isometry [Geometric Group Theory]

How can I prove that if $S,S'$ are two different finite generating sets of a group $G$ , then the metric spaces induced by the "word metric" are quasi-isometric? The definition of quasi-isometry is: ...
4
votes
1answer
116 views

Free subgroup of an amalgamated free product

I have the following question: Let $G=A\underset{C}\star B$ be the free product of two groups $A$ and $B$ with amalgam $C$, such that $C\cap vCv^{-1}=1$ for all reduced words $v$ in G with length ...