Questions tagged [geometric-group-theory]
Geometric group theory is the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act. Consider using with the (group-theory) tag.
681
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How do group symmetries work exactly?
Currently in my discrete mathematics course at the university I study at, we are on the topic of permutations and groups. We have learned that a group $G$ can be expressed in terms of a set of ...
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When is the word metric on a finitely generated group right-invariant?
Let $G$ be a finitely generated group with finite generator $S$, and define the metric $ d_S : G \times G \to \mathbb{Z} \cap [0, \infty)$ by
$$d_S(x, y) = \min \left\{ \ell \geq 0 : \exists s_1, \...
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Prove that there is unique length metric on $X/G$ such that $\pi : X \rightarrow X/G$ is a local isometry
Suppose $X$ is a length metric space and $G$ acts on $X$ by isometries properly discontinuously and the action is free. Then there is unique length metric on $X/G$ such that $\pi : X \rightarrow X/...
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Proof of Uniqueness of Free Group [closed]
I just wanted to verify if my proof is correct as I have to present this proof in my next class.
Consider the comm. diagram drawn above. We can verify that it commutes due to the categorical ...
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1
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Motivation for Hyperbolic Groups - Soft Question
I took a Geometric Group Theory course this semester. A very big part was hyperbolic groups. What I felt was a little bit lacking in this course was - why do I need hyperbolic groups? What is the ...
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Residually nilpotent groups with no infinite subgroup that is nilpotent
Is there an example of an infinite, residually nilpotent, torsion group that does not contain an infinite abelian subgroup.
I am looking for particular example or a reasoning why such a group cannot ...
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Examples of hyperbolic and non-hyperbolic space for quasi-isometric spaces
Let $X$ and $Y$ are quasi-isometric spaces. I try to find an example for which one of these spaces will be hyperbolic, other is not hyperbolic.
I know that for geodesic metric space if one of the ...
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1
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27
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Local to global theorem for quasi isometry
I am looking for a local - to - global principle in quasi-isometry.
Suppose $Z$ and $X$ are proper, geodesic, hyperbolic metric spaces such there is a local quasi-isometry $f: Z \to X$. That it for ...
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Understanding a step in the proof of Solovay-Kitaev theorem
There is a step in the proof of the proof of Solovay-Kitaev theorem about the existence of a set containing words of at most length length $l_0$ that cover $SU(2)$ . The proof I'm reading in given in ...
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Conditionally negative definite functions on free groups
Let $G$ be a group. I call a function $\ell\colon G\to [0,\infty)$ a conditionally negative definite (cnd) length function if
$\ell(g)=0$ iff $g=e$,
$\ell(g)=\ell(g^{-1})$,
$\sum_{g,h\in G}\overline{\...
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Simple Closed Curve Under Dehn Filling. [closed]
Let us consider the figure eight knot($K$) complement in $S^3$. Let $\gamma$ be an essential simple closed loop inside the figure eight knot complements in $S^3$.Now thicken the knot and consider the ...
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Figure-Eight Knot Complements in $S^3$
Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise ...
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1
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Prove that certain group has exponential growth
Let $R$ be the ring of dyadic rationals. I'm trying to prove that
$$G = \left\{\begin{bmatrix}
2^n & 0 \\
r & 1
\end{bmatrix}, r \in R, n \in \mathbb{Z}\right\}$$
has exponential growth.
I've ...
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$xy^2x^{-1}=y^3$ implies $[xyx^{-1},y]=1$
I want to show that if $x,y$ are elements of an hyperbolic group $G$, and $xy^2x^{-1}=y^3$, then $[xyx^{-1},y]=1$.
I tried to show that by using triangle thinness but I reached nothing. Another ...
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Order of generators of commutator subgroup
Let $H$ be commutator subgroup of a free group $G$. This free group $G$ is generated by $a$ and $b$ such that both have infinite order. We know that $H$ is not finitely generated. However, can we ...
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1
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constructing the dense subset in SU(2)
I read a comment that any two elements in $SU(2)$ each with infinite order would generate a dense subset in $SU(2)$. (Proper and dense subgroup of $\mathrm{SU}(2)$ ). The following point was made in ...
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Subgroups of semidirect product and commuting automorphism
I am not sure if this question actually makes sense, so any corrections/suggestions would be greatly appreciated!
Let $K$ be a polycyclic group, and let $\phi_1$ and $\phi_2$ be two automorphisms in $...
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Quasi-isometry group of $\{n^3;n\in\mathbb{Z}\}$.
Exercise $5.E.6$ of Clara Löh's $\textit{Geometric Group Theory. An Introduction}$ asks me to determine the quasi-isometry group of $A=\{n^3 | n ∈ Z\}$. If I change that $n^3$ by, for example, $n!$, ...
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Permuting subgroups with the same finite index in finitely generated free abelian group
This is a follow-up to this question.
Let $H$ be a subgroup of $\mathbb{Z}^2$ with finite index $m$. Let $\phi$ be an automorphism on $\mathbb{Z}^2$. Then $\phi$ corresponds to a matrix $M$ in $ \...
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Cocompact action (of a discrete group) with finite stabilizer on a cell-complex is proper
Assume $G$ is a discrete group, acting cocompactly and cellularly on a cell-complex with finite stabilizers (of cells). Then why is the action proper?
It seems we can deduce the cell-complex is ...
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Automorphism of $\mathbb{Z}^2\rtimes\mathbb{Z}$
Let $M$ be a matrix in $ \operatorname{GL}(2, \mathbb{Z})$, are there any books/articles that give a description of $\operatorname{Aut}(\mathbb{Z}^2\rtimes_M\mathbb{Z})$?
According to this paper, the ...
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1
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Classify the growth of poly-$\mathbb{Z}$ group
Let $G$ be a poly-$\mathbb{Z}$ group, i.e $$G =(\dots((\mathbb{Z} \rtimes_{\phi_1} \mathbb{Z})\rtimes_{\phi_2} \mathbb{Z}) \rtimes_{\phi_3} \mathbb{Z} \dots ) \rtimes_{\phi_{n-1}} \mathbb{Z}$$ What ...
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Permuting subgroups with the same finite index in free abelian group
Let $H$ be a subgroup of $\mathbb{Z}^2$ with finite index $m$. Let $\phi$ be an automorphism on $\mathbb{Z}^2$. Then $\phi$ corresponds to a matrix in $ \operatorname{GL}(2, \mathbb{Z})$.
Question: ...
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1
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Use the idea of stacking diagrams to show that $S_n$ is generated by $\{(i \text{ } i+1):1\leq i \leq n-1\}$
Use the idea of stacking diagrams to show that $S_n$ is generated by $\{(i \text{ } i+1):1\leq i \leq n-1\}$.
In the book Office Hours with a Geometric Group Theorist, the authors give the following ...
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Tools for determining whether the given group is residually finite?
How can I determine the residual finiteness of finitely presented groups? In my case group given by three generators, two relators :
$G=\langle a,b,s| s^bs=b^2, b^2a=ab^2\rangle$ , where the ...
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1
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Does a group of infinite abelian rank necessarily have exponential growth?
The abelian rank of a group $G$ is the maximum $n$ such that $G$ contains an isomorphic copy of $\mathbb{Z}^n$. A group has infinite abelian rank if it contains $\mathbb{Z}^n$ for every $n$.
If $G$ is ...
- 6,522
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Does there exist a finitely generated, torsion group $G$ with a residually finite ascending HNN extension?
Let $G$ be a group with an injective endomorphism $\phi$, then the HNN-extension
$$G_\phi = \left<G,t \mid t^{-1} gt= \phi(g) \right> $$ is called the ascending HNN-extension of $G$ determined ...
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Cayley complex of $\langle a|a^2\rangle$ is a covering of $\mathbb{R}P^2$
In Hatcher Example 1.47, we constructed the $2$-fold cover of $\mathbb{R}P^2$ by finding the Cayley complex of $G=\langle a|a^2\rangle$. I understand the construction and how the $2$-cells are ...
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What are some math books which are technical and rigorous yet accessibly written?
One of my favorite books that I discovered recently is "Office Hours with a Geometric Group Theorist" which I find to be very enjoyable, written accessibly, but yet still with a rich amount ...
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Is there an algorithm to check that a subgroup of a CAT$(0)$ group is *not* quasiconvex?
Let $G$ be a finitely generated CAT$(0)$ group and $H$ a subgroup. If $H$ is quasiconvex then it is finitely generated, so we can immediately conclude that any non-finitely generated subgroup of $G$ ...
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Davis Regular Tessellations of Spheres and Straight Line Coxeter Groups
In Davis' "Geometry and Topology of Coxeter Groups", section B.3, in particular Theorem B.3.1, there is a proof that every finite "straight line" Coxeter group is associated to a ...
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1
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Generalization of Bass-Serre Theory for pro-trees, $\Lambda$-trees, and $\mathbb R$-trees
If a group $G$ acts on a tree $T$, then by Bass-Serre theory $G$ can be rewritten as the fundamental group of the graph of groups $G \backslash\!\backslash T$.
There are several approaches to ...
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Arithmetic Fuchsian groups -- equivalent definitions?
I am trying to learn about arithmetic Fuchsian groups from Svetlana Katok's book Fuchsian groups.
At the beginning of Chapter 5, it says that the Fuchsian group is arithmetic if it has integer ...
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2
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Example of groups that are not quasi-isometric but have the same growth rate?
I have started working on group growth earlier this year, mainly using Drutu and Kapovich's notes. This morning I found myself wondering if I could find an example of groups that are not quasi-...
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Question about example of CAT($k$) space
Good time of day. I have the following question. It's written in wiki in examples of CAT($k$) spaces (https://en.wikipedia.org/wiki/CAT(k)_space) that the closed subspace $X$ of $\mathbb E^3$ (where $\...
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Subgroups of hyperbolic 3-manifold groups
Let $M$ be a hyperbolic 3-manifold and $\Gamma$ a finitely generated
subgroup of $\pi_1(M)$. Then either $\Gamma$ is geometrically finite,
or there exists a finite cover $M'$ of $M$ which fibers over ...
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The analogy between surfaces and vector space
When I am reading the first chapter of "A Primer on Mapping Class Groups" by Farb and Margalit, there is a beautiful analogy between surfaces and vector spaces. I have interpreted as the ...
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Is there a group of self-biholomorphism up to homotopy or isotopy?
I am recently studying the mapping class group defined as $Mod(S)=Homeo^+(S,\partial S)/\sim$ where $\sim$ is an equivalence relation on homotopy/isotopy. However, I am wondering what if we change ...
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1
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Ultrametric spaces are $0$-hyperbolic
Let $(X, d)$ be an ultrametric space. In particular, X satisfies the strong triangle inequality: for any $x, y, z \in X$, we have $$d(x,y) \leq \max\{d(x,z), d(y,z)\}.$$
I want to show that $X$ ...
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Semi-direct product in Tao's proof of Gromov's theorem
Terrence Tao provided an elementary proof of Gromov's theorem (https://terrytao.wordpress.com/2010/02/18/a-proof-of-gromovs-theorem/). I have been working my way through the proof and am stuck at the ...
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The figure eight knot complement in $S^3$.
Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise ...
- 29
1
vote
1
answer
189
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Are there any usages for growth rate that are relatively easy to show?
I'm giving a lecture about growth rates in a seminar. the chapter in the book I'm working with mainly focuses on computing growth rates and states Gromov's polynomial growth theorem (and the fact that ...
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2
answers
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If a finitely generated semi-direct product $\mathbb{Z}$ acting on a non finitely generated group, can the fixed points form a non-normal subgroup?
Suppose that we have a finitely generated and residually finite group $G = K\rtimes\mathbb{Z}$, but $K$ is not finitely generated. Let $T$ be a finite subset of $K$ such that $\langle (0,1), (k,0)\mid ...
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What is the relationship between amenability and property (T)?
I'm viewing Chapter 10 of GTM276 which focuses on some properties of topological groups including amenability and property (T). A footnote says they are almost exclusive. What does it mean? Does it ...
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Reference request: proof of the Rips Construction
I'm trying to understand how the Rips Construction works.
In particular, I'd like to understand why the presentation cooked up by the Rips construction (which if I'm not mistaken is not explicitly ...
- 151
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0
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42
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Is every coarse map between proper geodesic spaces a quasi-isometric embedding?
Let $(S,d_{1})$ and $(S',d_{2})$ be two proper geodesic metric spaces. Is every coarse map $f: S\rightarrow S'$ a quasi-isometric embedding?.
Just to recall, a coarse map $f: S\rightarrow S'$ between ...
0
votes
2
answers
125
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Proving there exists a category corresponding to a group (Clara Loeh pg-14)
Context:
Definition 2.1.18 (Automorphism group). Let $C$ be a category and let $X$ be an object of $C$. Then the set ${\rm Aut}_C(X) $of all isomorphisms $ X \to X$ in $C$ is a group with respect to ...
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Remark after proof of existence of free groups (page-21,Clara loeh)
Context
Definition 2.2.4 (Free groups, universal property). Let $S$ be a set. A group $F$ containing $S$ is freely generated by $S$ if $F$ has the following universal property: For every group $G$ ...
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1
answer
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Details in existence of free groups proof (Clara Loeh,pg-22,23)
I have a few doubts in the proof of existence of free groups given by Clara Loeh in page-22,23 of her Geometric Group theory book.
Theorem 2.2.7: Let S be a set. Then there exists a group freely ...
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0
answers
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If $K\rtimes \mathbb{Z}$ is a finitely generated and resdiually finite group but $K$ isn't, can the following abelianization all be finite?
I am looking for a residually finite semidirect product with the following properties. This is related to this question: If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, can ...
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