# Questions tagged [geometric-group-theory]

Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations

576
questions

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votes

**1**answer

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### Non-cocompact action on CAT(0) cube complex and hyperplane stabilizers

A hyperplane of a cube complex $X$ is a connected component of taking an $(n-1)$-cube for each midcube of $X$ and identifiying midcubes along faces of adjacent $n$-cubes of $X$.
If a group $G$ acts ...

**1**

vote

**1**answer

51 views

### Parallel hyperplane to a hyperbolic isometry of a CAT(0)-cube complex

Consider a finite-dimensional geodesically complete CAT(0) cube complex $X$.
A hyperplane of $X$ is a convex subspace of $X$ that intersects the mid-point of edges of $X$ such that $X - H$ has two ...

**5**

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**0**answers

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### Finitely generated nilpotent groups as cusp groups

I recently learned about the following question, asked by I. Kapovich :
Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\geq 3$...

**4**

votes

**2**answers

225 views

### Are symplectomorphisms of Weil–Petersson symplectic form induced from surface diffeomorphisms?

Let $S$ be a closed hyperbolic surface of genus $g\geq 2$. Let $(\mathcal{T},\omega)$ be the corresponding Teichmuller space with the Weil–Petersson symplectic from $\omega$. Let $\Phi:\mathcal{T}\...

**10**

votes

**1**answer

158 views

### The rigidity of the countable product of free groups

For a natural number $n$ let $F_n$ be the free group with $n$ generators.
The group $F_n$ is endowed with the discrete topology.
Given an increasing sequence $\vec p=(p_k)_{k\in\omega}$ of prime ...

**5**

votes

**1**answer

178 views

### Curvature and asphericity of cube complexes

Let $K$ be a connected cube complex (one may assume that its a cellulation of a smooth, closed manifold). Such a $K$ comes equipped with a length metric (one assumes that each edge is of unit length). ...

**7**

votes

**1**answer

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### Hyperbolic groups and spaces of negative curvature

Mikhail Gromov states that he "tried for about 10 years to prove that every hyperbolic group is realizable by a space of negative curvature" in his interview with Martin Raussen and Christian Skau (...

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57 views

### Amenability of the group of outer automorphisms of a connected compact Lie group

Forgive my ignorance on the Lie theory. I have the following questions in my current work concerning a certain property of compact connected Lie groups.
First, allow me to fix some notations.
Let $G$ ...

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**0**answers

119 views

### Does $AA^{-1}$ have the unique product property?

Let $A$ be a finite subset of a torsion free group $G$ with $|A|\geq3$. Does the set $AA^{-1}$ have the "unique product" property (i.e. there exist an element $c\in AA^{-1}$ that is uniquely written ...

**3**

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85 views

### Methods for constructing or checking for nontrivial classes in de Rham cohomology with local coefficients

Let $M$ be a smooth manifold (possibly with boundary), $E \to M$ a flat vector bundle, and $\mathcal{L}$ the corresponding sheaf of parallel sections.
Given a de Rham cohomology class $[\omega] \in H^...

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votes

**2**answers

506 views

### Is an HNN extension of a virtually torsion-free group virtually torsion-free?

This is a cross post from Math.StackExchange after 2 weeks without an answer and a bounty being placed on the question.
Let $G=\langle X\ |\ R\rangle$ be a (finitely presented) virtually torsion-free ...

**4**

votes

**0**answers

79 views

### Two definitions of horofunction for Gromov hyperbolic spaces

Let $X$ be a proper, geodesic, $\delta$-hyperbolic metric space (e.g. a hyperbolic group), and let $x_0$ be a basepoint for $X$. There seem to be two different definitions of "horofunction" for $X$, ...

**-1**

votes

**1**answer

154 views

### Groups With Arbitrarily Large Torsion [closed]

Thompson's Group has two well known presentations:
$\langle x_0,x_1, ... $ | $ x_k^{-1} x_n x_k = x_{n+1}\forall k < n \rangle$
$\langle A,B $ | $ [AB^{-1}, A^{-1}BA], [AB^{-1}, A^{-2}BA^2] \...

**5**

votes

**1**answer

176 views

### “Dimension” of discrete subgroups of infinite covolume in Lie groups

Let $G$ be a semisimple Lie group with finite center, $K$ a maximal compact
subgroup, and $d=\dim(G/K)$. Let $\Gamma$ be a non-cocompact discrete subgroup of $G$. [Edit: assume that $\Gamma$ is ...

**2**

votes

**1**answer

93 views

### Ends of G-spaces with action of a finitely generated group

This question is a development of my previous question.
Let $G$ be a finitely generated group acting transitively on an infinite set $X$ so that for every $g\in G$ and $x\in X$ the $g$-orbit $\{g^nx:...