# Tagged Questions

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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### Wallpaper groups for the hyperbolic plane

I would be grateful if someone could direct me to a reference that classifies the equivalent of the wallpaper groups (and the frieze groups and the point groups, if possible) for the hyperbolic plane, ...
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### What is the group of rotations of a volleyball(pyritohedron)?

Practice test for Abstract algebra final, very stuck on this particular question.
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### Solvable group which is not Virtually nilpotent

What is a example of Solvable group which is not Virtually nilpotent(does not have any nilpotent subgroup of finite index)?
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### Find the rank and the free generators

Consider the homomorphism$\$ $f:\ F\{x,y\} \to <x,y|x^2, y^3, xyx^{-1}=y^{-1}>$, find the free generators of $kerf$. I know that we should first consider the wedge sum of circles whose ...
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### If $G$ is two-ended finitely generated group, than $G$ is virtually $\mathbb{Z}$

I am trying to prove that every two ended finitely generated group is virtually $\mathbb{Z}$. My first idea is to find an element $g\in G$ such that $\mathbb{Z} = <g>$. $e(G) = 2$, so there is ...
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### Is the Cayley graph of a word-hyperbolic group a CAT(0) metric space?

It is mentioned on the Wikipedia article for Hadamard spaces that the Cayley graphs of a word-hyperbolic (f.g.) group are CAT(0) metric spaces. Is it so? My question comes from the fact that the ...
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### Examples of infinite Semi-direct products

I'm looking for some examples of semi-direct products, $G = N \rtimes_\alpha H$ of (infinite) groups. I'm aware of the definitions involved but never really thought through a lot of examples. I would ...
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### A problem in hyperbolic group

Let $G = \langle S \mid R \rangle$ be $\delta$-hyperbolic, and and $x$ is a word with minimal length such that $g_1,g_2 \in G$ with $g_1 = xg_2x^{−1}$, then why do we have: ...
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### explicit equivalent relation in the expression of the classifying space of a monoid

Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. (It's also called the internal nerve.) The ...
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### Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?

Let $f,g: \mathbb R \to \mathbb R$ be the permutations defined by $f: x \mapsto x+1$ and $g: x \mapsto x^3$, or maybe even have $g:x \mapsto x^p$, $p$ an odd prime. In the book, by Pierre de la ...
May I refer you to theorem $8.32$ on page $146$ in Metric Spaces of Non-Positive Curvature In the last paragraph, why is it the case that $H$ has finite index implies there is a constant $\mu$ such ...