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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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        16 views

        Hilbert space compression of lamplighter over lamplighter groups

        $C_2 \wr \mathbb{Z}$ is the lamplighter group but I'm currently looking at the lamplighter group with this group as a base space. Question: Consider the group $C_2 \wr (C_2 \wr \mathbb{Z})$, what is ...
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        105 views

        Simple One-relator Groups [on hold]

        One-relator groups, that is, groups which admit a finite presentation $\langle A \: | \: w=1 \rangle$ for some $w \in A^\ast$, are well studied objects in combinatorial group theory. Many abstract ...
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        148 views

        Which groups contain a comb?

        The comb is the undirected simple graph with nodes $\mathbb{N} \times \mathbb{N}$ where $\mathbb{N} \ni 0$ and edges $$ \{\{(m,n), (m,n+1)\}, \{(m,0), (m+1,0)\} \;|\; m \in \mathbb{N}, n \in \mathbb{N}...
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        99 views

        Uniform versus non-uniform group stability

        Group stability considers the question of whether "almost"-homomorphisms are "close to" true homomorphisms. Here, "almost" and "close to" are made rigorous using a group metric. More precisely, ...
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        votes
        1answer
        78 views

        Criterion for visuality of hyperbolic spaces

        I am trying to understand the following sentence on p. 156 of Buyalo-Schroeder, Elements of asymptotic geometry: "Every cobounded, hyperbolic, proper, geodesic space is certainly visual." Let $X$ be ...
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        1answer
        233 views

        Is residual finiteness a quasi isometry invariant for f.g. groups?

        A "residually finite group" is group for which the intersection of all finite index subgroups is trivial. Suppose $G$ and $G'$ are two quasi-isometric finitely generated groups. Does the residual ...
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        107 views

        Minimum Simple Burger-Mozes Type Group

        Burger and Mozes constructed (Burger and Mozes - Lattices in products of trees) infinite, finitely presented, torsion-free simple groups which split as amalgams of two finitely generated free groups ...
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        1answer
        71 views

        Non-collapsible complexes

        I don't know much about simple homotopy theory, so maybe my question is quite trivial: How does one prove that some complexes are contractible but non-collapsible? there are few refinements of this ...
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        153 views

        Breuer-Guralnick-Kantor conjecture and infinite 3/2-generated groups

        A group $G$ is called $\frac{3}{2}$-generated if every non-trivial element is contained in a generating pair, i.e. $$\forall g \in G \setminus \{e \}, \ \exists g' \in G \text{ such that } \langle g,g'...
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        1answer
        217 views

        Quasimorphisms and Bounded Cohomology: Quantitative Version?

        Consider maps from a discrete group $\Gamma$ to the additive group $\mathbb{R}$. A function $f:\Gamma \to \mathbb{R}$ is called a quasimorphism if it is locally close to being a group homomorphism. ...
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        1answer
        186 views

        Decidability: Presentations vs. Groups

        This question is just a curiosity for me as a non-expert. Quite often we ask about decidability of various properties in a group. Often the answer is that the property is undecidable in general. ...
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        92 views

        Second bounded cohomology and normal subgroups

        It may be a naive question, but: If a finitely generated group has an infinite-dimensional second bounded cohomology group, does it imply that it contains "many" normal subgroups? But "many", ...
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        2answers
        264 views

        Bieberbach theorem for compact, flat Riemannian orbifolds

        In his thesis, Bieberbach solved Hilbert 18 problem and proved that any compact, flat Riemannian manifold is a quotient of a torus. I need a reference to an orbifold version of this result: any ...
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        1answer
        157 views

        Second Bounded Cohomology of a Group: Interpretations

        Suppose we have a group $\Gamma$ acting on an abelian group $V$. Then it is well-known that the second cohomology group $H^2(\Gamma,V)$ corresponds to equivalence classes of central extensions of $\...
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        106 views

        Word length norm in the symmetric group $\mathfrak{S}_r$

        Consider on the symmetric group $\mathfrak{S}_r$ the generating system $\{\tau_i;\,1\le i\le r-1\}$ with $\tau_i = \langle i,i+1\rangle$ and the corresponding word length norm $N$. Now let $\tau\in\...

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        山西福彩快乐十分钟
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