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Questions tagged [geometric-group-theory]

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.

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. One can consider finitely generated groups themselves as geometric objects via Cayley graphs and the word metric. This leads to the study of large-scale invariants of metric spaces, where the local structure is essentially ignored.

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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 surjective. They use Arzela-Ascoli theorem for…
GGT
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Lipschitz retractions on finitely generated groups

Let $G$ be a finitely generated group and let $|\cdot|$ be a word norm (word length function) on $G$ with respect to some finite generating set. For every $n\in\mathbb{N}$, let $B_n$ denote the ball of radius $n$ around $1_G$, i.e. the set $\{g\in…
user446046
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Splitting groups over subgroups

Let $G$ be a finitely presented group with a subgroup $H$(if it helps we can assume that $H$ is finitely presented as well.) Is there any method in order to check that whether $G$ splits over the subgroup $H$ or not?
Bobby
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Composition of isometries of trees which is loxodromic(hyperbolic) with nonempty intersection of axis.

Question: If $\gamma,\delta\in$ Isom $T$ and Min$(\gamma)\cap$ Min$(\delta)=\emptyset$, then $\gamma\delta$ is loxodromic, Min$(\gamma\delta)\cap$ Min $(\gamma)\neq\emptyset$, and Min$(\gamma\delta)\cap$Min$(\delta)\neq\emptyset$. What I know: part…
S. Phanzu
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Which of these spaces below are quasi-isometric?

Question: Determine whether the following spaces are quasi-isometric: $\mathbb{H}^m$, $\mathbb{H}^n$, trees, $\mathbb{R}^m$, $\mathbb{R}^n$ with $m\neq n$. What I know: $\mathbb{H}^m$, $\mathbb{H}^n$ and trees are quasi-isometry as they belong to…
C. Thapa
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The fundamental group of graphs of groups

My question is about the computation the fundamental group of graph of groups: First let me give a reference for my question: In the page 14 of Groups acting on graphs by Dunwoody and Dicks. It says that The fundamental group of graphs of groups can…
Jivid
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Geometric Group Theory: cobounded orbit implies cocompact action

I'm reading the geometric group Theory notes by Bowditch. Assume X is a complete, locally compact geodesic space. A group of isometries G acts on X properly discontinuously. Then X/G is Hausdorff, complete and locally compact. Suppose some orbit is…
Han
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How to reconstruct a connected Lie group, from a given lattice in it?

I know that some works of Hillel Furstenberg and George D. Mostow, are in this direction. Can some one hint me towards the process of this reconstruction or send a link of the original papers, please.
user34942
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$G$-tree and $G$-subtree

Let $G$ be a finitely generated group acting on a locally finite tree $T$ such that each edge stablizer is finite. Is it possible to find a subtree $T'$ of $T$ in such a way that $G$ acts on $T'$ and the orbit graph $G/T'$ is finite? For instance,…
Bobby
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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.
user10
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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~
nflswsykimi
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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 groups are automatic and for other things. It is only…
mathreader
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#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 word length metric associated to the standard…
Mai Sasaki
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$\delta$ hyperbolic geodesic metric spaces

I have a basic question about the definition of $\delta$ hyperbolic geodesic metric spaces using triangles (studied in geometric group theory cours). The definition I studied in class is that a geodesic metric space is $\delta$ hyperbolic if for…
Dvir Cohen
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Proper discontinuous,co-compact action on locally finite CW-complex

Let $G$ be a group that acts properly discontinuously, co-compactly on a locally finite CW-complex $Z$. Does there exist a metric on $Z$ such that $Z$ becomes a geodesic metric space with that metric, and the action of $G$ becomes geometric (proper…
Yogi
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