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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Local isometry between non-positively curved cube complexes

Let $X$, $Y$ be non-positively curved cube complexes and suppose there is a local isometry $X \to Y$. If $Y$ is special, then so is $X$. This is Exercise 4.32 in M. Sageev's notes "CAT(0) Cube ...
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The compactness of metric space $\mathcal{G}_n$

Here, metric space $\mathcal{G}_n$ is described below: It is said that it is well-known that $\mathcal{G}_n$ is compact for every $n$. But I can't find a proof, can you give me a proof or an ...
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Some questions about a proof referring to hyperbolic group

I feel confused about: 1:It says that "otherwise H contains an infinite cyclic characteristic subgroup C," however, by definition, if H is elementary, we can only get that H contains an infinite ...
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Group action and orbit space

Suppose some group, G, acts on a space, X. Then an orbit of some $x\in X$ is defined as $$G.x = \lbrace g.x \mid g\in G\rbrace$$ Now consider the orbit space, $X/G$, the set of all orbits. I'm finding ...
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For a regular polygon with sides of length $l$, prove that all points within $l$ from a vertex lie on an incident edge

I am trying to prove that all the isometries of a regular polygon that map the polygon back onto itself must map vertices to vertices. I nearly have the proof, but I need to prove one more statement: ...
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Kazhdan's property (T) vs residual finiteness

There is a theorem that states that a discrete group $G$ with Kazhdan's Property $(T)$ and Property $(F)$ (so called factorisation property) is residually finite (see Kirchberg, Discrete groups with ...
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bounded generation and groups with infinitely many ends

Following section 7.1 in Peterson-Thom's paper here, we say a countable group $G$ is boundedly generated by the subgroups $G_1, \cdots, G_n$, if there exists an integer $k\in\mathbb{N}$, such that ...
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groups with infinitely many ends are not boundedly generated?

Recall that a group $G$ is boundedly generated if it can be written as a finite product of cyclic subgroups. And there are a lot of examples of groups that are (not) boundedly generated. I am ...
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71 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, ...
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What is the group of rotations of a volleyball(pyritohedron)?

Practice test for Abstract algebra final, very stuck on this particular question.
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50 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 ...
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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 ...
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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 ...
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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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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: $|(x_1...x_i)^{−1}g_1(x_1.....
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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 a\...
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``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" ...
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A question of quasi-isometric of free groups [closed]

For two free groups with finite ranks, are they quasi-isometric to each other?
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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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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 ...
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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: $|B(0,n)|=\frac{(2n)^k}{k!}+\frac{(2n)^{k-1}}{(...
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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 g=$a_1b_1......
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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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47 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 $$SL_2(\...
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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 $A$...
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50 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 ("...
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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 $f:\Omega\to\...
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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 $$\...
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1answer
119 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 ...
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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. ...
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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 ...
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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 $...
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59 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. ...
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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?
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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.
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1answer
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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 ...
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1answer
76 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) ...
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1answer
48 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 ...
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1answer
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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 ...
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111 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 ...
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155 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 ...
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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 V(T),\,\,G_v=\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...