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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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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30 views

A question about group actions on a trees [closed]

why does the following conclusion hold: Let G be a group acting on a tree $\Gamma$, H a subgroup of G with minimal subtree $\Gamma_H$ and $g\in G$ be a hyperbolic element, s.t. $\langle g\rangle\cap ...
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19 views

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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43 views

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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18 views

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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31 views

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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67 views

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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28 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 ...
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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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23 views

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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71 views

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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72 views

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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73 views

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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38 views

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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106 views

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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50 views

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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59 views

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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1answer
66 views

(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 ...
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2answers
89 views

Fundamental group of a closed hyperbolic surface is Gromov hyperbolic

Does anyone have a reference for the proof of the result in the title? Thanks!
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72 views

Fundamental domains of infinite-index subgroups of $SL(2,\mathbb{Z})$

While discussing modular forms associated to different subgroups $\Gamma$ of $SL(2,\mathbb{Z})$, there appeared to be a heuristic relationship between the index $[SL(2,\mathbb{Z}) \colon \Gamma]$ and ...
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1answer
25 views

Clarification from a proof that a certain type of graph can be endowed with a group operation

I need some help sorting out a construction of a group out of the vertices of a digraph with a certain property. I'll just throw some definitions here first... Definitions. An alphabet ...
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42 views

Group extensions, Geometric group theory

I am trying to learn a little bit of geometric group theory from the online lecture notes by Cornelia Drutu and Michael Kapovich titled geometric group theory. Here is the link ...
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28 views

Union of two disjoint triangles

First i must apologize for the bad english coming up, Im not used to writing math in english. If T1 is a regular triangle with its points A,B,C above the x axis and s0(T1) is the mirror image under ...
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an example of a non asynchronously combable group?

In several works of Martin Bridson there is a definition of an asynchronously combable group. It looks like this is a rather convenient notion used for estimating Dehn functions, for proving that ...
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1answer
68 views

Is every finitely generated Kleinian group commensurable to a Coxeter group?

Or, to a finitely generated reflection group? Here, I do not insist that the Coxeter group is represented as a hyperbolic reflection group. If not, what is an counterexample? And what is a ...
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1answer
36 views

Automorphism of Tree

Let $\sigma$ and $\theta$ be two automorphisms of tree $X$. I want to show that min$_{v\in V(X)}d(v,\sigma(v))=$min$_{v\in V(X)}d(\theta^{-1}\sigma\theta(v),v)$. I know every automorphism of tree is ...
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1answer
37 views

Question about particular words in the free group on three generators

In the free group generated by the letters $x,y,z$ suppose that we have a word such that for any one of $x,y,z$ the indices of each occurrence of that letter in our word sum to zero. Suppose further ...
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2answers
68 views

Cocompact group actions have cobounded orbits

Assume $X$ is a complete, locally compact, geodesic metric space (in particular, $X$ has closed balls are compact, by the Hopf-Rinow Theorem). Assume $G$ acts isometrically on $X$. We say $Q\subset X$ ...
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48 views

Let Cay(G, S) be the cayley graph of G with respect to the finite generating set S where G=⟨S∣R⟩ and R is finite.

Let $\operatorname{Cay}(G, S)$ be the cayley graph of $G$ with respect to the finite generating set $S$ where $G = \langle S\mid R\rangle$ and $R$ is finite. I am reading some notes that claim that ...
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125 views

Any subgroup of an abelian group is undistorted.

I need some help with the following math problem. I am studying some notes on Geometric Group Theory and I came across the following problem. Prove: Any subgroup of an abelian group is ...
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53 views

If $H$ is a subgroup of $G$, then the inclusion is a quasi-isometry if and only if $H$ has finite index in $G$.

If $H$ is a subgroup of $G$, then the inclusion is a quasi-isometry if and only if $H$ has finite index in $G$. I am reading through some notes on Geometric Group Theory and I came across this side ...
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1answer
49 views

What does it mean for a group to act cocompactly by isometries on a topological space $X$?

What does it mean for a group to act cocompactly by isometries on a topological space $X$? I know if $X$ is a topological group, and $A$ a subspace, then $A$ is cocompact iff $X/A$ is compact. Not ...
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52 views

Suppose that $X$ is locally compact and $G$ acting on $X$ is proper. Show that the quotient $X/G$ is Hausdorff.

Suppose that $X$ is locally compact and $G$ acting on $X$ is proper. Show that the quotient $X/G$ is Hausdorff. I am working through some notes on Geometric Group Theory and I am having a hard time ...
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59 views

Is a $0$-hyperbolic group free?

In his article, Abderezak Ould Houcine asks the following question: If $G$ is a hyperbolic group, let $\delta_0(G)$ denote the infinimum of $\delta$ for which $G$ is $\delta$-hyperbolic. When ...
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1answer
89 views

Free product as automorphism group of graph

Let $A$ and $B$ be two groups. We define following graph $X$. The set of vertices is the left cosets $gA$ and $gB$ where $g\in A*B$ (By $A*B$, I mean the free product of $A$ and $B$). The edges of the ...
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1answer
97 views

Trivial elements in $T(a,b,c)$

Consider the group $T(a,b,2)=<x,y|x^a, y^b, (xy)^2>$ and assume none of $a$ or $b$ is equal to $2$. How can one list all the trivial words (say up to length $11$ and apart from $(xy)^{2n})$) in ...
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1answer
62 views

Representation of cell location in hyperbolic plane

I want to represent an order-5 square tiling (image from Wikipedia; more text below image): Obviously for a simple grid I can uniquely refer to a given square by its ...
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1answer
67 views

General introduction to orbifolds?

Where should I go to learn about orbifolds? I am interested in a general introduction that gives precise definitions and clear explanations. I have a fair background in topological and smooth ...
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1answer
64 views

Fundamental domain of a Fuchsian group which is not locally finite

I am trying to understand Example 9.2.5 in Beardon's book The Geometry of Discrete Groups. The goal is to construct a fundamental domain of a Fuchsian group which is not locally finite. Definitions ...
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1answer
29 views

Growth of direct products

Let $G$ and $H$ be finitely generated groups. Is it true that $ \gamma_{G \times H} $ is weakly equivalent to $ \gamma_{G}\gamma_{H} $? Assuming the answer is yes, can one use exactly the same ...
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1answer
37 views

Are free products of exponential growth?

Let $G$ and $H$ be finitely generated groups. Is the free product $G*H$ always of exponential growth? I am guessing that the answer is yes, but I don't know how to prove it correctly. My idea is to ...
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1answer
41 views

Question about Lamination for free groups

I have a question about the leaves of "Stable lamination" as defined in the paper "LAMINATIONS, TREES, AND IRREDUCIBLE AUTOMORPHISMS OF FREE GROUPS" and the periodic conjugacy classes. More ...
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245 views

Showing that a group with a presentation is free/not free

Show that the group with presentation $\langle a, b, c \mid a^2cb^3\rangle$ is free with basis $\{a, b \}$. Show that the group with presentation $\langle a, b, c \mid a^3b^3 \rangle$ is not free. ...
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1answer
66 views

Showing that $\mathbb{R}\times [0,1]$ is quasi-isometric to $\mathbb{R}$

I have a question that asks me to show that $\mathbb{R}\times [0,1]$ is quasi-isometric to $\mathbb{R}$ I have having trouble showing what I have is a quasi isometry. My map is simply: ...
3
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1answer
63 views

Constructing The Cayley Graph and quasi-isometry to $\mathbb{Z}$

If we have a group $G$ defined by: $G=\langle a,b\mid b^2=1\rangle$ then I first need to construct the cayley graph of this, now I think that this is going to look like the "telephone pole" metric ...
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51 views

Showing that triangles in $\mathbb{Z}$ are thin

If we let $\mathbb{Z}$ be generated by $\{3,5\}$ then I have a question which asks me to show that geodesic triangles are $k$-thin and to find a minimum bound on $k$. I have been thinking about this ...
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1answer
42 views

$T_3$ is quasi isometric to $T_4$

I have a question which asks me to show that $T_3$ is quasi isometric to $T_4$, that is the three and 4 valence trees. I know that this means that I have to define a map $f:T_3\rightarrow T_4$ such ...
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1answer
47 views

Cayley graph of an amalgam

I was wondering what the Cayley graph of an amalgam $A*B$ (over $\{ 1\}$ ) is? $A, B$ are finitely generated. Can't work it out. Thanks!