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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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        4
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        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 ...
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        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 ...
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        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 $\geq 3$...
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        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}\...
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        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). ...
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        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 (...
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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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        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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        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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        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 ...
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        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 $X$, ...
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        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:...

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