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# 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

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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 ...
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### 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 ...
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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 $\geq 3$...
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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 $\Phi:\mathcal{T}\... 1answer 158 views ### The rigidity of the countable product of free groups For a natural number$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}$of prime ... 1answer 178 views ### Curvature and asphericity of cube complexes Let$K$be a connected cube complex (one may assume that its a cellulation of a smooth, closed manifold). Such a$K$comes equipped with a length metric (one assumes that each edge is of unit length). ... 1answer 116 views ### Hyperbolic groups and spaces of negative curvature Mikhail Gromov states that he "tried for about 10 years to prove that every hyperbolic group is realizable by a space of negative curvature" in his interview with Martin Raussen and Christian Skau (... 0answers 57 views ### Amenability of the group of outer automorphisms of a connected compact Lie group Forgive my ignorance on the Lie theory. I have the following questions in my current work concerning a certain property of compact connected Lie groups. First, allow me to fix some notations. Let$G$... 0answers 119 views ### Does$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}$that is uniquely written ... 0answers 85 views ### Methods for constructing or checking for nontrivial classes in de Rham cohomology with local coefficients Let$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^...
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### 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 ...
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### 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 $X$, ...
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Thompson's Group has two well known presentations: $\langle x_0,x_1, ...$ | $x_k^{-1} x_n x_k = x_{n+1}\forall k < n \rangle$ $\langle A,B$ | $[AB^{-1}, A^{-1}BA], [AB^{-1}, A^{-2}BA^2] \... 1answer 176 views ### “Dimension” of discrete subgroups of infinite covolume in Lie groups Let$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$is ... 1answer 93 views ### Ends of G-spaces with action of a finitely generated group This question is a development of my previous question. Let$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:...

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