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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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 ...
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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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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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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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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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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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28 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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57 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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36 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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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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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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40 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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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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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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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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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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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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45 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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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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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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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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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 ...
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Simplicial complex and link

Theorem 7.16 in Bridson and Haefliger's book states that: Theorem: Let K be an $M_k$-simplicial complex, and let $x\in K $. If the number $\epsilon(x)$ defined in (7.8) is strictly positive, ...
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Limit sets of Kleinian groups

Just wondering if there is some good reference (textbooks or expository papers) on Kleinian groups and limit sets? Thanks~
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Proper action and compactness

Proposition: Suppose a group $\Gamma$ acts properly by isometries on a metric space $X$. If the action is cocompact then every element of $\Gamma$ is a semisimple isometry of $X$. (Please refer ...
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on Cayley diagrams

is the picture the Cayley Graph of the group $\langle a,b,c\mid a^2, b^2,c^2\rangle$ ? What would it be for $\langle a,b,c\mid a^2b^2c^2\rangle$?
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Topological Space with Given Fundamental Group

We know that if we want to construct a space with a given fundamental group $G$ ,we can use cells and attaching maps, or fundamental domains and attaching maps, as in : How to determine space with a ...
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Conditions for finiteness of group in geometric group theory

Are there any sufficient conditions in geometric group theory for a group to be finite? Are there any necessary conditions?
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Exercise 1.1 in Serre's trees

I have in fact become stuck by the very first problem in Serre's book on Trees. It is a little bit embarrassing but ho-hum. I start with Serre's definition of direct limits. Let $(G_i)_{i \in I}$ be ...
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Which Words Are Part of a Free Basis of $F_n$?

Start with a free group on $n$ generators, $F=\langle a_1,\ldots, a_n\rangle$. If I write a word, $w$, in these generators, is there an efficient algorithm to determine whether or not $w$ can be ...
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Not all finitely-presented groups are fundamental groups of closed 3-manifolds

It is a well-known result that, for any finitely-presented group $G$ and any integer $n \geq 4$, there exists a closed $n$-manifold whose fundamental group is isomorphic to $G$ (a sketch of proof can ...
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(compact, non-empty boundary )Surface Geodesics on Hyperbolic Geometry

I have a basic knowledge of hyperbolic geometry . I am trying to understand the meaning of "a compact surface S with non-empty boundary(which is neither a disk nor an annulus )with a complete ...