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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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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N-filling implies 3N/2 - geodesic filling

Suppose X is a geodesic space and c is a rectifiable loop.Show that if c admits an N-filling then c admits a 3N/2- geodesic filling. I suspect that there is nothing to prove indeed but still I ...
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63 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 ...
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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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35 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 ...
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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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24 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 ...
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1answer
58 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 ...
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28 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 ...
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23 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 ...
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1answer
29 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 $X$?
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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 \ ...
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35 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 ...
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1answer
31 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 ...
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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 ...
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75 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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23 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 ...
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1answer
30 views

“Uniqueness” of the Multi- Dehn-twist

I'm trying to writing down a proof for the following claim about Dehn-Twists: Let $\{a_1,...,a_m\}$ be a collection of distinct nontrivial isotopy classes of simple closed curves in a surface $S$ ...
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1answer
60 views

Is my graph a tree?

Let M be a smooth connected manifold. G is a group act on M cocompactly and suppose there is a harmonic function $h$ on M with minimal energy.$h:\rightarrow [0,1]$ such that h is nonconstant and ...
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hyperbolic isometry

I have a project I have to do. In order to do it I need to investigate this book. In page 94 they defined hyperbolic isometry on a metric s.t it possesses no fixed point in the tree. After that they ...
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1answer
42 views

Horosphere of a metric tree

I have a project I have to do. In order to do it I need to investigate W.E. Grosso's translation of "The Green Book" on Hyperbolic Group Theory, as found here. I try to understand the term horosphere ...
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(Un)distorted subgroups in $\mathbb{F}_2 \times \mathbb{F}_2$

We consider the product of free groups $$\mathbb{F}_2 \times \mathbb{F}_2 = \langle a,b,c,d \mid [a,b]=[b,c]=[c,d]=[d,a]=1 \rangle.$$ Given some elements $g_1,\ldots,g_n \in \mathbb{F}_2 \times ...
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77 views

Integers and perfect cubes are isometric

Is there a quick way to check if the integers and $\{n^{3}: n \in \mathbb{Z}\}$ are quasi-isometric? It seems that they are since they look the "same" from far away.
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1answer
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Number of path connected components

Is it possible to give an explicit characterization of compact subsets of $[0,\infty)$. Is it true that given any compact subset $K \subseteq [0,\infty)$ then $[0,\infty) \setminus K$ has only one ...
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1answer
42 views

Integers are geodesic

Consider $\mathbb{Z}$ the integers. Is this considered a geodesic metric space under the usual metric? Why? given $n,m$ integers, we would require to have a map $f: [0,c] \rightarrow \mathbb{Z}$ such ...
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87 views

Examples of connections between bounded cohomology and geometric properties of groups

I think that the question is self explanatory. The only example that comes to my mind is the characterization of hyperbolic groups given by Mineyev ("Straightening and bounded cohomology"). There is ...
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Action of $Homeo(S)$ on $Out(\pi_1(S))$ for $S$ orientable surface

Let $S$ an orientable surface of any genus, possibly punctured and with boundary: with the classical notation $S=S_{g,n}^{b}$ where $g$ is the genus, $n$ the number of punctures and $b$ the numer of ...
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A small cancellation group does not contain $\mathbb{Z}^3$

I read somewhere that a small cancellation group (ie. a group admitting a presentation statisfying the small cancellation condition $C'(1/6)$) does not contain $\mathbb{Z}^3$, but without a precise ...
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Identification of polygon edges

In Klein's famous example of regular 14-gon made of 336 copies of (2,3,7) triangles, he used identification for edges such that side 2i+1 is identified with side 2i+6 (mod 14). But I wonder how could ...
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1answer
69 views

Decomposition of a group whose Cayley graph is a tree

This is an exercise taken from Chapter 9 of a French book, Géométrie et Théorie des Groupes. It says, roughly, the following: Show that a finitely generated hyperbolic group, whose Cayley graph is a ...
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1answer
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Is $Out(F_n)$ of type FP$_\infty$

Is $Out(F_n)$ of type $FP_\infty$? $Out(F_n)$ acts on Outer space $X_n$ which is finite dimensional, locally finite and contractible. However the action is only proper (stabilizers of points ...
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Sequence of bounded sequence in metric space

I am reading a paper and bumped at this lemma which I do not know the proof and would like to see some reference. Please suggested me a possible reference. Let $M$ be a metric space and ...
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142 views

Are all $\delta$-hyperbolic groups CAT(0)?

In Alessandro Sisto's notes on geometric group theory he mentions that "Many, probably most people in the field" believe that not all $\delta$-hyperbolic groups are CAT(0) groups. Can anything be said ...
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harmonic function on manifold

Let M be a 2 dimensional manifold. $h:M\rightarrow R$ be a harmonic function from manifold to real line. G is group that act by isometry. $g*h(x)=h(g(x))$. Let $W=\{x|h(x)=t\}$ that is the level set ...
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1answer
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Are right-angled Artin groups always CAT(0)?

To each right-angled Artin group $A_\Gamma$ there is an associated space $S_\Gamma$ on which the group acts on (the Salvetti complex). The fundamental group of the Salvetti complex is the right-angled ...
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Gradient curve of a harmonic function

I am reading the paper "Energy of Harmonic function and Gromov proof of Stalling theorem" https://www.math.ucdavis.edu/~kapovich/EPR/energy.pdf I have no clue about the lemma 8.4(i). What is gradient ...
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1answer
52 views

Is $\mathbb{Z}_3$ CAT(0) and/or (Gromov) $\delta$-hyperbolic?

This example is confusing me. Is $\mathbb{Z}_3 = \langle a\vert a^3\rangle$ $\operatorname{CAT}(0)$ and/or (Gromov) $\delta$-hyperbolic? The Cayley graph clearly has bounded diameter, therfore it is ...
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1answer
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Ultralimit of Cayley graph of $\mathbb{Z}^2$

I am new to ultralimits and I am trying to find out what the asymptotic cone $\operatorname{Cone}_{\omega}(X)$ of $X:=\operatorname{Cay}(\langle\mathbb{Z}^2\vert(1,0),(0,1)\rangle)$ is. And how to ...
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71 views

Geometric Interpretation of S3

My impression was that the symmetric group $S_3$ acts on the vertices of a labeled triangle. However, I am not sure this is the case anymore, because of the following. (The triangle is labeled as ...
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Group G having two ends

Let $Z$ set of integer is a subgroup of G. $Ends(G)=2$. Is it true that $Z$ is of finite index in G ? I was trying to show quasi isometry between $Z$ and $G$. For notation and other definition ...
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one end group with positve first Betti number$\beta^{(2)}_1(G)>0$

Could anyone give me an example of a countable finitely generated (f.g.) discrete group $G$ with one end but have non-trivial $H^1(G,\ell^2G)$? To be precise, consider the following two cases. (1) ...
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Arzela-Ascoli in proper metric space.

In the book by Bridson and Haefliger http://www.math.bgu.ac.il/~barakw/rigidity/bh.pdf page 145. lemma 8.28 To prove part 2 of the lemma that is The natural map for $G_{x_0}\rightarrow Ends(X)$ is ...
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1answer
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Any infinite property (T) subgroup of $Aut(F_n)$?

I heard that it is still an open problem whether $Aut(F_n), n\geq 4$ has Kazhdan's property (T), where $F_n$ denotes the non-abelian free group on $n$ generators, my question is: Does there exist any ...
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convex distance function from a point in CAT(0) space.

Let the metric space (X,d) be a CAT(0) space with the metric d. I am trying to show that the function $d_p(x)=d(p,x)$ where p is fixed point and x varies is a convex function by using the definition ...
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When is the automorphism group of the Cayley graph of $G$ just $G$?

Let $G$ be a finite group and $S$ a generating set of $G$. We can draw the Cayley graph $C(G,S)$ by putting each element of $G$ as a vertex, and drawing an edge between two elements $g$, $h\in G$ iff ...
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Cone over a circle.

Let K be a cone over a cirlce $C$ of Length L.That is $K=C\times[0,\infty)/(C,0)$ it is normal euclidean cone. In notation of Bridson Haefliger here is a link ...
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Whats the difference between modular forms of different levels?

We have a natural surjective group homomorphism: $\phi : SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/(n\mathbb{Z}))$ from which, given any subgroup $H<SL_2(\mathbb{Z}/(n\mathbb{Z}))$, we may take the ...
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1answer
31 views

Shapes of a simplicial complex

In Bridson and Haefliger's book, page 98, there is a definition of shape. Here is a link to the book. The definiton is not very clear to me. It says set of isometry classes: Is it the isometry ...