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
4
votes
1answer
67 views
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
1answer
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
votes
0answers
83 views
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
2answers
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
1answer
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
1answer
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
1answer
116 views
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 (...
3
votes
0answers
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 $G$ ...
1
vote
0answers
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
votes
0answers
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^...
15
votes
2answers
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
0answers
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
1answer
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
1answer
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
1answer
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:...
