# 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.

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### The analogy between surfaces and vector space

When I am reading the first chapter of "A Primer on Mapping Class Groups" by Farb and Margalit, there is a beautiful analogy between surfaces and vector spaces. I have interpreted as the ...
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### Is there a group of self-biholomorphism up to homotopy or isotopy?

I am recently studying the mapping class group defined as $Mod(S)=Homeo^+(S,\partial S)/\sim$ where $\sim$ is an equivalence relation on homotopy/isotopy. However, I am wondering what if we change ...
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1 vote
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### Ultrametric spaces are $0$-hyperbolic

Let $(X, d)$ be an ultrametric space. In particular, X satisfies the strong triangle inequality: for any $x, y, z \in X$, we have $$d(x,y) \leq \max\{d(x,z), d(y,z)\}.$$ I want to show that $X$ ...
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### Semi-direct product in Tao's proof of Gromov's theorem

Terrence Tao provided an elementary proof of Gromov's theorem (https://terrytao.wordpress.com/2010/02/18/a-proof-of-gromovs-theorem/). I have been working my way through the proof and am stuck at the ...
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### The figure eight knot complement in $S^3$.

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise ...
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### Are there any usages for growth rate that are relatively easy to show?

I'm giving a lecture about growth rates in a seminar. the chapter in the book I'm working with mainly focuses on computing growth rates and states Gromov's polynomial growth theorem (and the fact that ...
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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 ...
1 vote
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### Proper and cocompact group action by isometries on a metric space

I am reading the book "Metric spaces of Non-Positive Curvature" by Bridson and Haefilger and got stuck with the following : Proposition II.6.10(2) Let $\Gamma$ be a group that acts properly ...
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### Does $A$ have infinite order in $G= \langle A,B \ |\ B A B^{-1} = A^2 \rangle$?

I have a group (arising from the fundamental group of a manifold) $$G= \langle A,B \ |\ B A B^{-1} = A^2\rangle$$ and I would like to show that $A$ is an element of infinite order inside $G$. ...
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### Do there exists conditions we can put on two groups which have the same growth rate, so that their Cayley graphs are isomorphic?

Given a finitely generated group $G$ with a generating set $S$, we can define the growth rate function of a group, denote it $\#_{G,S}(n)$. It is clear that two groups having the same growth rate ...
1 vote
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### Reference for solvability of Baumslag-Solitar groups [duplicate]

I've been searching for a while and couldn't find any reference which provides necessary and sufficient conditions over $m,n$ for the group $BS(m,n)$ to be solvable. The only I could find is that it ...
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### Torsion free groups with no unique products (notation)

I am reading a paper by William Carter titled "New examples of torsion-free non-unique product groups" and saw the following group: P_k=\langle a,b\mid ab^{2^k}a^{-1}b^{2^k},ba^{2}b^{-1}a^{...
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1 vote
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### Commensurable, Quasi-Isometric and Finitely Generated Groups

Metric Spaces of Non-Positive Curvature, Book by André Haefliger and Martin Bridson, page $141$. Two groups are said to be Commensurable if they contain isomorphic subgroups of finite index. ...
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### The homomorphisms $\Gamma\to\mathbb{R}$ are linear combinations of the functions $f_g$
I am reading Brooks and Series, Bounded Cohomology for Surface Groups. At the end of the first paragraph on the last page, it says "the homomorphisms $\Gamma\to\mathbb{R}$ are linear combinations ...