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
987 questions
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The ten most fundamental topics in geometric group theory
What are the ten most fundamental topics in geometric group theory?
This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
8
votes
1
answer
223
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Preserving non-conjugacy of loxodromic isometries in a Dehn filling
Suppose that $g$ and $h$ are non-conjugate loxodromic isometries in a cusped hyperbolic $3$-manifold $M$ of finite volume. Fix a cusp $T$ of $M$. Can I choose a hyperbolic Dehn filling of $M$ along $...
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Visual boundary vs Bowditch boundary [closed]
Is there any difference between the visual boundary of a relatively hyperbolic group and the Bowditch boundary of a relatively hyperbolic group?
Visual boundary is generally associated to a CAT(0) ...
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3
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1
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$\mathrm{SL}(2,\mathbb{Z})$ finitely generated by using the Schwarz-Milnor lemma
Recently, I have been studying the modular group $G=\mathrm{SL}(2,\mathbb{Z})$, and I am trying to prove $G$ is finitely generated by using the Schwarz-Milnor lemma in geometric group theory.I am ...
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Is there a Cayley graph with end space infinite and discrete?
A Cayley graph of a finitely generated group must be locally finite, and we know end spaces of locally finite graphs must be compact - so we can't have an infinite and discrete end space in this ...
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12
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Does every mapping class group embed into some $\mathrm{Out}(F_n)$?
The title is pretty much the whole question. Let $S_g$ be a closed, oriented surface of genus $g$. Does there exist $n$ such that the mapping class group $\mathrm{Mod}(S_g)$ embeds as a subgroup of $\...
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The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results.
Let $A$ be a finitely generated abelian group,...
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4
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Estimating an upper bound of hyperbolicity constants in Gromov-hyperbolic groups
Is there any good references with an explicit estimating hyperbolicity constants of hyperbolic groups? I was thinking about whether there is any relation between the relator of maximal length in the ...
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8
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Finite two-relator groups and quotients of knot groups
Let $G$ be a one-relator group $\langle A \mid R = 1 \rangle$. Then clearly $G$ is finite if and only if it is cyclic of finite order, i.e. can be given by a presentation $\langle a \mid a^n = 1 \...
8
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$K$-theory and its dual
I am reading a paper which uses some $K$-homology which is the homology theory dual to $K$-theory can be defined using the homotopy theoretic formulation:
$$
K_\ast(X)\cong\pi_\ast(K\wedge X).
$$
...
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5
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Groups with a weaker (?) form of the Helly property
Let $G$ be a finitely generated group, with a fixed word metric $d$ coming from some finite generating set. Fix $n\in \mathbb{N}$, and suppose that for all finite $S\subseteq G$ with $\mathrm{diam}(S)=...
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Krasner–Kaloujnine universal embedding theorem for finitely generated groups?
The Krasner–Kaloujnine universal embedding theorem states that any group extension of a group $H$ by a group $A$ is isomorphic to a subgroup of the regular wreath product $A \operatorname{Wr} H$. When ...
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3
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1
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100
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Representation of Lie groups inducing a quasi-isometric embedding of their symmetric spaces
Let $G_{1}$ and $G_{2}$ be connected semisimple real Lie groups with no compact factors and finite center and let $K_{1}$ and $K_{2}$ denote some fixed choice of their maximal compact subgroups, ...
4
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1
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Howson's property for amalgams of free groups
Recall that a group has the Howson property (also known as the finitely generated intersection property) if the intersection of any two finitely generated subgroups is again finitely generated.
I am ...
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Polynomial isoperimetric inequalities for finitely presented subdirect products of limit groups
Does every finitely presented subdirect product of limit groups admit a polynomial isoperimetric inequality? That is, does there exist a constant $C > 0$ and an integer $d \geq 1$ such that for ...
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