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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Closed $SL_2(\mathbb{Z})$ conjugacy class [on hold]

For what matrices $A \in SL_2(\mathbb{R})$ is the conjugacy class by $SL_2(\mathbb{Z})$ closed ?
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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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56 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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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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70 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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24 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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93 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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49 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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50 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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57 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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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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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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23 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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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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62 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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34 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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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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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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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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117 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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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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44 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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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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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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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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95 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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59 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
65 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
55 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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27 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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34 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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39 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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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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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: ...
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59 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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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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41 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
45 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!
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Dehn presentation question

I have just shown that if a group $G$ admits a Dehn presentation then there are finitely many conjugacy classes of finite order. I'm then trying to deduce from that fact that there is some $N$ such ...
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69 views

Baumslag–Solitar $B(1,2)$ is not hyperbolic

I have a question which asks me to show that the Baumslag–Solitar $B(1,2)$ is not hyperbolic by considering its Cayley graph and showing that triangles can be arbitrarily fat. The Cayley graph can be ...
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1answer
136 views

Applications of Cayley Graphs in Physics

I have been recently reading about Cayley graphs and character theory. It is evident that Cayley graphs are very useful tool in theoretical computer science. In physics, Cayley graphs seem do appear ...
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61 views

Quasi-isometry of Schreier coset graphs

Let $G$ be a finitely generated group and $H$ a subgroup. For any choice of a generating set $X$ we can form the Schreier coset graph. Is this graph independent of the generating set up to ...
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73 views

3-Dimensional analogue of a Dihedral Group

I'm reading about dihedral groups right now, and I'm being asked to find the orders of various groups of rigid motions of 3-dimensional polygons. What I want to know, though, is if there's a nice ...
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Is the fundamental group of a compact manifold finitely presented?

Let $X$ be a connected compact smooth manifold. If $X$ is boundaryless, we can choose a Riemannian metric for $X$ so that $\pi_1(X)$ acts geometrically (ie. properly, cocompactly, isometries) on the ...
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36 views

How to find linear fractional transformations of a group

Finite presentation of G is $$\langle x,y,t,q : x^2=y^3=t^2=q^2 =1,tq=qt,ty=yt,qyq=y^{−1},xt =qx \rangle.$$ I am interested in finding linear fractional transformations x,y,t,q which satisfy all its ...
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Does there exists a finitely presented group with Dehn function > n^3 and all asymptotic cones simply connected [closed]

it is well known that all asymptotic cones simply connected implies polynomial Dehn function (Gromov). It is also well known that quadratic Dehn function implies all asymptotic cones simply connected ...
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138 views

Intuition behind the failure of unimodularity

If $G$ is a locally compact group then up to normalization it admits a unique Haar measure: a left invariant measure defined on all Borel subsets of $G$, which assigns every compact set a finite ...
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2answers
111 views

Fundamental group of a Cayley graph

Could someone please indicate the proof of the following fact (I believe strongly, but I am not sure, that this is true). Let $F$ be a free group of finite rank, $N<F$ a normal subgroup and ...