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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50 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 ...
2
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1answer
49 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 ...
2
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1answer
85 views
Show the existence of a certain subgroup of F2
This is a homework question: I need to find three subgroups of the free group with two generators, $F_2$, with certain properties. I have found the other two by constructing covering spaces of $S^1 ...
2
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1answer
109 views
Seifert manifolds and Fuchsian group
A Fuchsian group is a discrete subgroup of $PSL(2, \mathbb{R})$.
Let $M$ be a Seifert manifold (maybe with boundary) and $t \in \pi_1(M)$ the class of a regular Seifert fiber.
Hempel claims in his ...
2
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1answer
157 views
Subgroup of a virtually cyclic group
Let $G$ be a virtually cyclic group, i.e., G has an infinite cyclic subgroup $H$ of finite index. Is it true that if $H'$ is another infinite cyclic subgroup of $G$ then $H'$ must be of finite index ...
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35 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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1answer
47 views
Splitting of a surface group over the subgroup associated to a closed geodesic
Let $G=\pi_1 (S)$ for a closed surface $S$, consider a closed geodesic $c$ on $S$ and let $H$ be the subgroup of $G$ induced by $c$. Is it true that $G$ splits over $H$, i.e. $G$ can be written as a ...
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403 views
How strong is the statement that Thompson F is amenable?
Justin Moore's proof turned out to have an error
I just attended Justin Moore's talk on this today. Since I am neither a group theorist nor a combinatorist, and is not familiar with ultrafilters I ...
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128 views
How useful are geometric aspects when studying finite groups?
My newbie impression when studying finite group theory is that geometric aspects are not very prevalent. Cayley graphs play quite some role for visualising finite groups, but compared to the study of ...
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104 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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134 views
Higman group with 3 generators is trivial
The group $G$ generated by $x,y,z$ subject to the relations $[x,y]=y$, $[y,z]=z$, $[z,x]=x$ is trivial. This isn't the case for the corresponding group with 4 generators, which is the famous Higman ...
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192 views
Combinatorial total space for finitely generated torsion-free groups?
Motivation: I'm an operator algebraist and I'm looking for an answer to the main question in order to build non-trivial spectral triples for a class (as large as possible) of discrete groups.
$\to$ ...
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151 views
Is there an analogue of outer Space to study outer automorphisms of free pro-$p$ groups?
I would like to know if there is an analogue of Culler & Vogtmann's outer space to study outer automorphisms of free pro-$p$ groups. Perhaps an initial guess of such a space would be a moduli ...
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180 views
Reference: Geometric group theory
H. Bass has studied existence of lattices on trees. Can someone suggest a (readable) reference for lattices on graphs?
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38 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 ...
3
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65 views
Classification of Bieberbach groups
Does anybody know if there exists a list of the four dimensional Bieberbach groups presented by generators and relations on the web?. I know there exists the book Crystallographic Groups of ...
3
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34 views
How to reconstruct a connected Lie group, from a given lattice in it?
I know that some woks of Hillel Furstenberg and Mostow, are in this direction. Can some one hint me towrds the process of this reconstruction or send a link of the original papers, please.
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54 views
The universal space of a Coxeter group
Consider a Coxeter group $(W,S)$ and a topological space $X$. We define a mirror structure on $X$ as a locally finite family $(X_{s})_{s\in S}$ of closed subspaces of $X$. Let's consider $W$ with the ...
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246 views
Double Coverings of the Double Torus
I'm trying to count all the double coverings of the double torus. I know that the fundamental group of the double torus is
$$\pi_1(X)=\langle a,b,c,d;[a,b][c,d]\rangle $$
where ...
3
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76 views
Construct tiling group from hyperbolic polygon
Given a hyperbolic $4n$-gon $P$ in the Poincaré disk, how can we construct explicitly the subgroup $G < \mathrm{Aut}{\left(\mathbb{D}\right)}$ which gives a tiling of $\mathbb D$ with fundamental ...
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34 views
#B(e,n) of $\mathbb{Z}^k$
notation:: #A is the number of factors of A, B(e,n)={x $\in$A|d(e,x)≦n}, and S(e,n)={x $\in$A|d(e,x)=n}
Then, I want to know that #B(e,n) of $\mathbb{Z}^k$.
Where $\mathbb{Z}^k$ is equipped with the ...
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37 views
Reference request for an explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface
Let $\Sigma_g$ be a geuns $g$ Riemann surface with $g \geq 2$.
It can be thought of in the following way:
it is the quotient space
$$\mathbb{H}/\pi_1(\Sigma_g)$$ where an element of ...
2
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75 views
The program “gm” by Epstein and Rumsby for drawing tesselations and Cayley graphs
In the book
Word Processing in Groups by David Epstein, there is a pair of pages, 38 and 39, which have two pictures on them. If you are familiar with the book, you probably know exactly what I am ...
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66 views
Sufficient conditions for quasi-isometric embeddings of Cayley graph
I would like to know more about the assumptions under which the Cayley graph of a given group embeds quasi-isometrically into the space where the group is acting.
For instance, if a group $G$ acts by ...
2
votes
0answers
100 views
Subgroups of amalgamated free product
My question is the following:
Suppose that we are given the amalgamated product $ G = G_1 * _{G_3} G_2 $, and subgroups $ H_i \le G_i $ for $i=1,2,3$, such that in addition $H_3$ is as large as ...
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50 views
A problem about normalizers in $PSL(V)$
Let $K=\mathbb F_{p^k}$ a finite field, and $V$ a vector space on $K$. Clearly $PSL(V)=SL(V)/SL(V)\cap Z$ acts on $V$ by the following rule ($Z$ is the subgroup of the scalar functions):
...
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58 views
is a free cellular action of a discrete group over a cellular complex always properly discontinuous?
I think that the answer is "yes" for free simplicial actions over simplicial complexes: a non trivial element g of the group G must map a simplicial simplex to a different one, because of brower ...
2
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32 views
maximal independent sets and maximal independent generating sets of PSL(2,p)
I'm searching for an example of a group PSL(2,p) with a "maximal independent" set of length greater than the length of a "maximal independent generating" set. All authors refer to Whiston's papers ...
2
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0answers
67 views
Groups acting on (regular) trees with finite quotient
Let $T$ be a regular tree, and suppose that $G \leq \mathrm{Aut}(T)$ has finite quotient graph, $T / G$. Is it true (in general) that $G$ will have trivial centralizer in the full automorphism group? ...
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33 views
Generators of SL_n of a local ring of integers.
This is a follow-up to my previous question: Generators of $GL_n(\Bbb Z)$ and $GL_n(\Bbb Z_p)$
Let $\mathcal{O}_K$ be the ring of integers in a characteristic zero non-archimedean local field $K$.
...
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60 views
JSJ-decompositions of groups and 3-manifolds: a reference request
I am, for whatever reason, interested in learning about the JSJ-decomposition of groups. Having asked around a bit, it was suggested I first learn about what is happening in the manifolds and then ...
2
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0answers
53 views
Dehn Twist in the sense of graphs
Does anyone knows a good book or script about Dehn Twists in the sense of graphs. More precisely: I need to know how a Dehn Twist yields an automorphism of a group or subgroups. I want to know ...
1
vote
0answers
50 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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55 views
Does there exists a finitely presented group with Dehn function > n^3 and all asymptotic cones simply connected
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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41 views
Ring of invariants for the action of rotation groups in tensors.
Consider the component-wise action of the group $SO(p)\times SO(q)$ in the tensor product of two real vector spaces $S^2(R^p)\otimes R^q$. How to parametrize orbits of this action ? For $q=1$ we ...
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57 views
Crystallographic Groups Question
A Crystallographic Group is a discrete group of isometries acting on the n-dimensional euclidean space $\mathbb{R} ^n $ with compact fundamental domain.
A lattice is a crystallographic group which ...
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0answers
195 views
Wreath Product of Two Finitely Generated Groups
Let $G$ and $H$ be two finitely generated groups, and let $W = G \wr H$ be the wreath product of $G$ and $H$. Show that $W$ is finitely generated. In class today, we were showed this and told that it ...
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172 views
Group acting on graph
If $S $ is infinite, locally finite graph which is not tree, $\tilde{S} $ is its universal cover, $p:\tilde{S}\rightarrow S $ is covering map,and $G $ is acting on $\tilde{S} $ with finite quotient $Y ...
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0answers
42 views
free subgroup of amalgamated free product groups
Suppose $G=A*_{C}B$ be an amalgamated free product group, in general, I am interested in the question whether $G$ contains a free subgroup, and given specific examples, I always first calculate its ...
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45 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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0answers
31 views
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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0answers
55 views
Quotient group and graphs
what is the Quotient group and how we can compute it for Petersen graph?
what properties of graphs are incurred in the quotient groups of graphs?
for example suppose G=(V,E) , D is the free abelian ...
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58 views
Infinite geodesic rays leaving a K-quasiconvex subgroup stay K-close to it.
I am going through some basic properties of $\delta$-hyperbolic spaces and groups and I am having some difficulties proving precisey some things that are anyway intuitively clear to me.
Let $G$ be a ...
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107 views
How can I generate $\mathrm{SL}(n,\mathbb Z)$ by the subgroup $\mathrm{SL}(n-1,\mathbb Z)$ and another Element of $\mathrm{SL}(n,\mathbb Z)$?
Let $\{z_1,...,z_n\}$ be the canonical Basis of $\mathbb{Z}^n$, such that $z_i$ equals the vector $(0,\dotsc,0,i,0,\dotsc,0)$ with a 1 in the $i$th position.
I want to show that the ...
