# 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.

6 views

### Local isometry between non-positively curved cube complexes

Let $X$, $Y$ be non-positively curved cube complexes and suppose there is a local isometry $X \to Y$. If $Y$ is special, then so is $X$. This is Exercise 4.32 in M. Sageev's notes "CAT(0) Cube ...
48 views

### The compactness of metric space $\mathcal{G}_n$

Here, metric space $\mathcal{G}_n$ is described below: It is said that it is well-known that $\mathcal{G}_n$ is compact for every $n$. But I can't find a proof, can you give me a proof or an ...
54 views

### Some questions about a proof referring to hyperbolic group

I feel confused about: 1:It says that "otherwise H contains an infinite cyclic characteristic subgroup C," however, by definition, if H is elementary, we can only get that H contains an infinite ...
78 views

### Group action and orbit space

Suppose some group, G, acts on a space, X. Then an orbit of some $x\in X$ is defined as $$G.x = \lbrace g.x \mid g\in G\rbrace$$ Now consider the orbit space, $X/G$, the set of all orbits. I'm finding ...
36 views

### For a regular polygon with sides of length $l$, prove that all points within $l$ from a vertex lie on an incident edge

I am trying to prove that all the isometries of a regular polygon that map the polygon back onto itself must map vertices to vertices. I nearly have the proof, but I need to prove one more statement: ...
70 views

### Kazhdan's property (T) vs residual finiteness

There is a theorem that states that a discrete group $G$ with Kazhdan's Property $(T)$ and Property $(F)$ (so called factorisation property) is residually finite (see Kirchberg, Discrete groups with ...
58 views

### bounded generation and groups with infinitely many ends

Following section 7.1 in Peterson-Thom's paper here, we say a countable group $G$ is boundedly generated by the subgroups $G_1, \cdots, G_n$, if there exists an integer $k\in\mathbb{N}$, such that ...
70 views

### groups with infinitely many ends are not boundedly generated?

Recall that a group $G$ is boundedly generated if it can be written as a finite product of cyclic subgroups. And there are a lot of examples of groups that are (not) boundedly generated. I am ...
71 views

### 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, ...
47 views

### What is the group of rotations of a volleyball(pyritohedron)?

Practice test for Abstract algebra final, very stuck on this particular question.
50 views

### 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 ...
53 views

### 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 ...
75 views

### 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 ...
56 views

### 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 ...
30 views

48 views

### angle" between two group elements

For the group $\mathbb{Z}^n$, we may embed them in $\mathbb{R}^n$ and then it is clear that for any two elements in $\mathbb{Z}^n$, we may treat them as vectors and hence the notion of angle" ...
41 views

### A question of quasi-isometric of free groups [closed]

For two free groups with finite ranks, are they quasi-isometric to each other?
42 views

### The definition of the number of ends for a locally finite graph

Here I am wonder what's the definition in terms of vector spaces over Z2, and how to show it's equivalent to other definitions.
59 views

### Hyperbolic groups from Dehn functions

Hyperbolic groups may be defined as finitely generated groups admitting a linear Dehn function. I wonder whether it is possible to prove most of the classifical properties of hyperbolic groups in this ...
46 views

51 views

### The conjugacy problem of finitely generated free group

I would like references for algorithms solving the conjugacy problem in $F_n$ (the free group on $n$ generators)?
47 views

119 views

### Relatively hyperbolic groups

A group $G$ is relatively hyperbolic relative to a collection of subgroups $\mathcal{G}$, if $G$ admits an action on a connected graph, $K$, with the following properties: (1) $K$ is hyperbolic, and ...
36 views

### Finitely generated group acting cocompactly on a Manifold of bounded geometry

Want a justification of the following fact. If G is a finitely generated group, then there exist a Riemannian manifold $M$ such that the action of $G$ is cocompact isometric properly discontinuous. ...
If $A$ and $B$ are groups we have the following short exact sequence: $$0 \to [A,B] \to A * B \to A \times B \to 0,$$ where the group $[A,B]$ is free (see e.g. Serre's Trees). I am wondering if ...