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.

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5
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1answer
61 views

Wallpaper groups for the hyperbolic plane

I would be grateful if someone could direct me to a reference that classifies the equivalent of the wallpaper groups (and the frieze groups and the point groups, if possible) for the hyperbolic plane, ...
3
votes
1answer
42 views

What is the group of rotations of a volleyball(pyritohedron)?

Practice test for Abstract algebra final, very stuck on this particular question.
-1
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0answers
34 views

Solvable group which is not Virtually nilpotent

What is a example of Solvable group which is not Virtually nilpotent(does not have any nilpotent subgroup of finite index)?
2
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1answer
46 views

Find the rank and the free generators

Consider the homomorphism$ \ $ $f:\ F\{x,y\} \to <x,y|x^2, y^3, xyx^{-1}=y^{-1}>$, find the free generators of $kerf$. I know that we should first consider the wedge sum of circles whose ...
1
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0answers
41 views

If $G$ is two-ended finitely generated group, than $G$ is virtually $\mathbb{Z}$

I am trying to prove that every two ended finitely generated group is virtually $\mathbb{Z}$. My first idea is to find an element $g\in G$ such that $\mathbb{Z} = <g>$. $e(G) = 2$, so there is ...
2
votes
1answer
61 views

Is the Cayley graph of a word-hyperbolic group a CAT(0) metric space?

It is mentioned on the Wikipedia article for Hadamard spaces that the Cayley graphs of a word-hyperbolic (f.g.) group are CAT(0) metric spaces. Is it so? My question comes from the fact that the ...
2
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3answers
50 views

Examples of infinite Semi-direct products

I'm looking for some examples of semi-direct products, $G = N \rtimes_\alpha H$ of (infinite) groups. I'm aware of the definitions involved but never really thought through a lot of examples. I would ...
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0answers
24 views

A problem in hyperbolic group

Let $G = \langle S \mid R \rangle$ be $\delta$-hyperbolic, and and $x$ is a word with minimal length such that $g_1,g_2 \in G$ with $g_1 = xg_2x^{−1}$, then why do we have: ...
4
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0answers
55 views

Are these groups solvable?

I am thinking of Baumslag-Solitar groups of type $BS(1,m)=\langle a,b \mid bab^{-1} = a^m\rangle$ as a prototype. We can think of them as follows: Start with an infinite cyclic group $\langle ...
1
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0answers
46 views

``angle" between two group elements

For the group $\mathbb{Z}^n$, we may embed them in $\mathbb{R}^n$ and then it is clear that for any two elements in $\mathbb{Z}^n$, we may treat them as vectors and hence the notion of ``angle" ...
0
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1answer
35 views

A question of quasi-isometric of free groups [closed]

For two free groups with finite ranks, are they quasi-isometric to each other?
0
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1answer
40 views

The definition of the number of ends for a locally finite graph

Here I am wonder what's the definition in terms of vector spaces over Z2, and how to show it's equivalent to other definitions.
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0answers
52 views

Hyperbolic groups from Dehn functions

Hyperbolic groups may be defined as finitely generated groups admitting a linear Dehn function. I wonder whether it is possible to prove most of the classifical properties of hyperbolic groups in this ...
1
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1answer
43 views

Ball growth function for $Z^n$

in this book: https://issuu.com/yilongzhang/docs/piotr_w._nowak__guoliang_yu-large_s in the Example 3.1.9. it is said that for group ($Z^k$,+) we have: ...
-1
votes
2answers
65 views

The graph of free product group. [closed]

For the free product $A*B$=G, where $A$ and $B$ are groups, there is a graph defined by: the edge set E(G)$\backsimeq$G and the vertex set V(G)$\backsimeq$G $G/A \bigsqcup G/B$, and to ...
3
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1answer
49 views

The conjugacy problem of finitely generated free group

I would like references for algorithms solving the conjugacy problem in $F_n$ (the free group on $n$ generators)?
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0answers
43 views

Cayley Graph of $SL_2(\mathbb{Z})$

As the title suggest, I'm interested in depicting (if possible) the Cayley graph of the special linear group $SL_2(\mathbb{Z})$. I know one has to start with a presentation of the group so ...
2
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0answers
40 views

Realising Seifert-van Kampen in 2-complexes

Suppose you have a group $G$ given by a finite presentation $\langle X; \mathbf{r}\rangle$, and also suppose you know that this group decomposes as a free product with amalgamation $A\ast_CB$ (with ...
2
votes
1answer
48 views

Modify a Dehn presentation

Suppose you have a Dehn presentation $\langle X \mid R \rangle$ of (say not the free group) a hyperbolic group. Has there been some work done on changing this presentation, e.g. adding a relation ...
0
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0answers
79 views

problem with proof 'horoball QI extension theorem'

I'm reading the book of Drutu and Kapovich "Lectures on geometric group theory". In the proof of Mostow rigidity theorem, they say that they can extend an $\rho$-equivariant function ...
3
votes
3answers
86 views

Countable abelian groups are amenable.

I am using the following definition of amenable group: The at most countable group G is amenable iff there exists a sequence $\{F_n\}$ of finite subsets such that for every $g\in G$ we have ...
2
votes
1answer
116 views

Relatively hyperbolic groups

A group $G$ is relatively hyperbolic relative to a collection of subgroups $\mathcal{G}$, if $G$ admits an action on a connected graph, $K$, with the following properties: (1) $K$ is hyperbolic, and ...
2
votes
1answer
36 views

Finitely generated group acting cocompactly on a Manifold of bounded geometry

Want a justification of the following fact. If G is a finitely generated group, then there exist a Riemannian manifold $M$ such that the action of $G$ is cocompact isometric properly discontinuous. ...
2
votes
0answers
44 views

Short exact sequences for amalgameted free products and HNN Extensions

If $A$ and $B$ are groups we have the following short exact sequence: $$ 0 \to [A,B] \to A * B \to A \times B \to 0, $$ where the group $[A,B]$ is free (see e.g. Serre's Trees). I am wondering if ...
1
vote
1answer
35 views

Weakly relatively hyperbolic groups

A finitely-generated group $G$ is weakly hyperbolic relatively to a collection of subgroups $\{ H_1, \ldots, H_r\}$ if the graph obtained from a Cayley graph of $G$ by coning off the cosets of the ...
2
votes
1answer
53 views

Centralizers in mapping class groups

According to Nielsen-Thurston classification, given a closed surface $S$, the elements of the mapping class group $\mathrm{MCG}(S)$ lie in three categories: periodic, reducible and pseudo-Anosov. ...
2
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0answers
53 views

Classification of Torus bundle over $\mathbb{S}^1$ [closed]

$T^2$ has a fixed free isometry group as $D_8$, is this true that $T^2$ bundle over $S^1$ have three type?
2
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0answers
44 views

Classification of closed flat three dimensional manifold? [closed]

Obviously $T^3$ is one type. $K^2\times S^1$ is also one type. Maybe $T^2\tilde{\times}\mathbb{S}^1$ bundle is also one type.
1
vote
1answer
29 views

Find the Cayley graph of the group $G = ( \mathbb{Z}/2 \mathbb{Z}) × ( \mathbb{Z}/2 \mathbb{Z})$ with generating set ${(1, 0),(0, 1)}$.

I am quite new to geometric group theory and can't really visualize the way the Cayley graph changes when you change the generating set. I found many possible graphs here but I don't really see how to ...
4
votes
1answer
71 views

Infinite torsion CAT(0) groups

Do all infinite CAT(0) groups contain a $\mathbb{Z}$ subgroup? I am aware that this has been established for hyperbolic groups, and similar questions have appeared on open questions lists for CAT(0) ...
3
votes
1answer
47 views

Relative hyperbolicity and splittings

A subgroup $H \leq G$ is almost malnormal whenever $|H \cap H^g| <+ \infty$ for every $g \notin H$. For the definition of relatively hyperbolic groups, see for instance there. Do you know an ...
3
votes
1answer
68 views

Hyperbolic groups and cohomology

I'm looking for a reference (survey article, monograph, etc.) that treats the cohomological properties of hyperbolic groups and/or its generalizations as relatively hyperbolic groups, lacunary ...
2
votes
0answers
104 views

Linear Isoperimetric Inequality is invariant under quasi isometry

Suppose $X$ and $Y$ are quasi-isometric. Show that $X$ satisfies a linear isoperimetric inequality iff $Y$ satisfies a linear isoperimetric inequality. My idea: Suppose $X$ satisfies a linear ...
5
votes
2answers
137 views

Is every metric space quasi-isometric to a graph?

I've proved that if $(X, d)$ is a geodesic metric space then there exists a graph which is quasi-isometric to $X$. During this proof I've precisely used the fact that given two point in $X$ there ...
2
votes
2answers
57 views

Tree of groups $(\mathcal{G},T)$

Let $G$ be act on $\Gamma$ with a fundamental domain $T$ where $T$ is tree. We construct tree of groups $(\mathcal{G},T)$ with the following structure: $$\text{for every } v\in ...
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0answers
20 views

Weyl (anti-)invariant differential operators on spheres

The permutation group $S_n$ acts as via Weyl reflections of $A_{n-1}$ on $R^{n-1}$ und thus on the sphere $S^{n-2}$. On this sphere, we have a natural action of $SO(n-1)$ generated by the angular ...
1
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2answers
61 views

Isometric group of hyperbolic 3-dim manifolds

In the book : Foundation of Hyperbolic Manifolds There is a theorem that any finite subgroup of $ Isom(\mathbb{E}^n) $ fixes a point. And I hope to solve the following question : Any ...
2
votes
1answer
54 views

Using CAT(0) inequality

Let $X$ be a CAT(0) space with metric $d$. Let $p,x,y$ three points on $X$, and let $u,v$ be points on geodesic $[p,x]$ and geodesic $[p,y]$ such that $d(p,u)\geq a,d(p,v)\geq a$,where a is some ...
0
votes
1answer
19 views

Maximal subset with finite Assouad-Nagata dimension

Given some space $X$ with non-finite Assouad-Nagata dimension. Is it possible for a subset $Y \subset X$ with finite Assouad-Nagata dimension to exist such that $Y$ is maximal in the sense that if any ...
2
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0answers
32 views

Can SAGE or othe software compute or guess growth rates of infinite discrete groups?

I am interested in the growth rate of some finitely generated (infinite, non-abelian) discrete groups. Knowing very little about geometric group theory, I am wondering if I can plug them into sage and ...
2
votes
1answer
73 views

Fuchsian group with parabolic element

I'm interested in this problem: let $\Gamma \subset PSL(2, \mathbb{R})$ a Fuchsian group (i.e. it is a discrete subgroup) which contains the trasnformation $\gamma \colon z \mapsto z+1$ then the ...
1
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0answers
34 views

What's the asymptotic of the radius of the Rubik square Cayley graph?

This post is a sequel of The Rubik Square permutation groups, which should be read first to understand the notation. Question: what's the radius$^*$ of the Cayley graph of $G_n$ generated by the red ...
0
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1answer
43 views

Definition of almost geodesic metric space

Is there a term for a metric space that satisfies the following condition: There exists a $C \geq 0$ such that for all $x,y \in X$ there exists a path $\gamma :[0,1] \to X$ with $\gamma(0)=x$ and ...
2
votes
1answer
46 views

action of a subgroup on a metric space

If $G$ acts properly and cocompactly by isometries on the metric space $X$ and if $H$ is a subgroup of $G$. Does $H$ act properly and cocompactly by isometries on a subspace of $X$?
4
votes
2answers
66 views

When is a right-angled Coxeter group one-ended?

Let $\Gamma$ be a simplicial graph (ie. without multiple edes nor loops). We define the associated right-angled Artin group $A(\Gamma)$ by the presentation $$\langle v \in V(\Gamma) \mid [u,v]=1 \ ...
4
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1answer
80 views

Geometric Representation of Quasidihedral Groups

I am going back through Dummit/Foote studying for a prelim and came across the 'quasidihedral' or 'semi-dihedral', group of order $2^n$, with presentation $\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs ...
2
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1answer
37 views

explicit equivalent relation in the expression of the classifying space of a monoid

Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. (It's also called the internal nerve.) The ...
16
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0answers
347 views

Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?

Let $f,g: \mathbb R \to \mathbb R$ be the permutations defined by $f: x \mapsto x+1$ and $g: x \mapsto x^3$, or maybe even have $g:x \mapsto x^p$, $p$ an odd prime. In the book, by Pierre de la ...
0
votes
1answer
80 views

The ends of a group

May I refer you to theorem $8.32$ on page $146$ in Metric Spaces of Non-Positive Curvature In the last paragraph, why is it the case that $H$ has finite index implies there is a constant $\mu$ such ...
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0answers
27 views

Class preserving Autpmorphism

This time I am having interest in theory of Class preserving Automorphism and central preserving Automorphism and some topics related to Automorphism of groups. So Could you please tell me some ...