# 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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### Quasi-isometric Classification of Free Products of Surface Groups

Let $S_g$ denote the compact surface with $g$ holes, and denote its fundamental group as \begin{equation} \Sigma_g=\pi_1(S_g)=\left<a_1,b_1...,a_g,b_g\Big| \prod_{ i \in \{1,...,g\}} [a_i,b_i]\...
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### Examples of non-free group actions on trees with finite edge-stabilizers

I am interested in finding examples of finitely-generated non-free groups $H$ such that $H$ is a finite index subgroup of some group $G$ and $H$ acts without edge-inversion on some tree $T$ with ...
4answers
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### An easy example of a non-quasiconvex subgroup

Let $G$ be a finitely generated group, and consider the surjection $\mu:F(A)\to G$ induced by the set of generators $A$, where $F(A)$ is the free group on $A$. A word $w$ is said to be ($\mu$-)...
1answer
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### Finite generation of vertex groups of a cyclic splitting of a hyperbolic group and generalisations of Grushko Theorem

Let $G$ be a finitely generated word hyperbolic group. Suppose $G$ acts non-trivially (without a global fixed point) on a tree without inversions and with cyclic edge stabilizers. Is it true that the ...
1answer
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### Seeking an example of a group with finite presentation for which the Word Problem is not solvable

In the book Geometric Group Theory of Clara Loh, it is proven that the Word Problem is solvable for hyperbolic groups. It is also stated that the Word Problem is not solvable in all finitely presented ...
1answer
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3answers
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### Interesting Theorems on Finitely Generated Abelian Groups?

First time teaching algebraic topology, probably gonna be related to most of my questions on here for a while. I was wondering if anyone knows of particularly interesting theorems or examples from the ...
1answer
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### Is $\mathbb{Z}$ Gromov-Hyperbolic?

If $G$ is a finitely-generated group, then we say that $G$ is Gromov-Hyperbolic if it's Cayley Graph, $\operatorname{Cay}(G, S)$, is a Gromov-Hyperbolic metric space. Now in the case of the group of ...
1answer
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### Group acting on trees with relations $abc=1$ and $cba=1$

Suppose $G$ is a finitely presented group that acts on a tree $T$ by isometries, and let $a,b,c\in G$ with relations $abc=1$ and $cba=1$. If two of $a,b,c$ are hyperbolic, does this imply the third ...
2answers
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