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.

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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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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 ...
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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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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 ...
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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 ...
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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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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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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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Given a finitely generated, residually finite, non-cohopfian group $G$, can the following abelianizations ALL be finite?

Let $G$ be a finitely generated, residually finite, non-cohopfian group. Since $G$ is residually finite, we know that there exists a sequence of nested, normal, finite index subgroups $$G = N_0 \rhd ...
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Is this semi-direct product residually finite?

Consider the group $G=K\rtimes \mathbb{Z}$ defined as follows: The subgroup $K$ is generated by elements $x_i,y_k$ with $i,k \in {\mathbb Z}$ and $k > 0$, and it has defining relations \begin{...
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Clarification on the Nielsen-Schreier Theorem

I am a new student of Geometric Group theory, and my professor walked us through a proof of the Nielsen-Schreier Theorem that uses the fact that a group that acts freely on a tree must be free. Our ...
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Permuting subgroups with the same finite index

Suppose that we have a finitely generated residually finite group $G = \langle g_1,\ldots,g_r \rangle$ and $H$ is a subgroup of $G$ with finite index $m$. Let $\phi$ be an automorphism on $G$. ...
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Spectral radius of a finitely generated group

Let $G$ be a finitely generated group and $\Gamma$ be its Cayley graph with the usual word metric. Let $\mu$ be a non-degenerate measure on $G$, and define $p(x, y) = \mu(x^{-1} y)$. As is well-known, ...
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Two fold mapping from $S^3$ to $SO(3)$

Let $S^3$ be a unit cube in $\mathbb{R}^3$ with the boundary points identified. Let $\chi$ be a continuous, surjective map from $S^3$ onto $SO(3)$ where the preimage $\chi^{-1}(x)$ has a cardinality ...
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Is the intersection of quasi convex sets also quasi convex?

A subset $A$ of a geodesic space $X$ is called quasi convex if there exists a constant $k > 0$ so that if $x,y \in A$, the geodesic joining $x$ to $y$ is in the $k$ neighborhood of $A$. Is it true ...
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If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, can the following abelianization be finite?

Suppose that we have a finitely generated group $G = K\rtimes\mathbb{Z}$, but $K$ is not finitely generated. Let $T$ be a finite subset of $K$ such that $\langle (0,1), (k,0)\mid k \in T \rangle$ is a ...
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non finitely generated group with finite abelianization

Suppose $K$ is a non finitely generated and residually finite group, is it possible that $K$ has finite abelianization, i.e. the quotient group $K \big/ [K,K]$ is finite? If we take $K$ to be the ...
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Tesselation of hyperbolic plane are hyperbolic

Let’s take a tesselation $T$ of the hyperbolic plane (not necessarily regular), my intuition tells me that clearly $T$ should be hyperbolic itself (in the sense of Gromov or using $\delta$-slim ...
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Existence of a transvection which maps one hyperplane to another.

Following screenshot is of a proof from the book Classical groups and geometric algebra by Larry C. Groove. I have some doubt in the proof. My question: How $\tau$ becomes a transvection on $V$? To ...
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Algebraic Characterisation of the End Space of a proper geodesic space in terms of non-continuous functions

Let $X$ be a rimcompact Tychonoff space, that is, a completely regular Hausdorff space with a base of open sets with compact boundaries. It is well known that $X$ has a maximal compactification with a ...
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A lattice acts in a locally compact group

I'm taking a topics course and trying to fill in details from a lecture. I'm hoping someone can help fill things in or point to a resource where these basics are covered. We have a lattice $\Lambda &...
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Computing how many elements of a finitely generated group has a certain word length, using software?

What I want is to give the generators and relations, as well as an integer $n$, then tell the program to compute the number of elements having word length $\leq n$. Can someone please show me how to ...
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8 votes
1 answer
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Are jumps in the growth function of an infinite group increasing?

Let $G$ be a group with a $S$ a finite subset of $G$ generating it, with $\{e\}\in S$ and $S=S^{-1}$, and let $\gamma_G^S$ be the growth function of $G$ respect to $S$, that is, $\gamma_G^S(l)$ is the ...
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How to understand the covering space of Mapping cylinder?

Let $\partial:E\rightarrow Y$ be a map between to topological spaces. Define $X$ to be the mapping cylinder $X=E\times [0,1] \bigsqcup Y / \sim : (x,1) \sim \partial(x), x\in E$. Then $X$ comes with a ...
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2 votes
1 answer
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Existence of a geodesic with endpoints at infinity

I want to prove the existence of a geodesic $\gamma$ with end points $\xi$ and $\xi'$ at infinity in a proper CAT(-1) space. A hint in Elements of Asymptotic Geometry is to first notice that each ...
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Floyd Compactification

Suppose $G$ is a finitely generated group, $f: \mathbb{N} \longrightarrow \mathbb{R_+}$ is a function satisfying $1<\frac{f(n)}{f(n+1)}<\lambda$ and $\sum f(n)<\infty$, which we called a ...
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A certain free product of groups is virtually torsion-free

Suppose that $G_1,\ldots,G_n$ are finite groups, and $m\geqslant 0$ is some integer. Set $$G=G_1\ast\cdots\ast G_n*F_m,$$ (where $F_m$ is the free group on $m$ generators). Then, is $G$ virtually ...
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Local geodesics

Suppose I have a $K$-local geodesic in a metric space $M$, $\alpha(t) : [0,L] \rightarrow M$, meaning $$\forall t ,t' \in [0,L] \quad \text{s.t.} \quad |t-t'|\leq K, \quad |\alpha(t)-\alpha(t')|=|t-t'|...
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question about group actions on metric spaces

Suppose we have a metric space $V$, a group $G$ and an action $\cdot: G \times V \rightarrow V$. What assumptions must I make so that the following is true? Claim: For each $x, y \in V$, if there ...
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7 votes
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Euclidean embeddings of cayley graphs

For any finite group $G$, we can define a 'dimension' of the group as the smallest $n$ such that there is some choice of generators $S$ of $G$ where $G$ equipped with word metric with respect to $S$ ...
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Borel quasi-isometry between proper geodesic spaces

Let $(S,d_{1})$ and $(S',d_{2})$ be two proper geodesic metric spaces. If there exists a quasi-isometric embedding $f: S\rightarrow S'$, does there exist a $\textbf{Borel}$ quasi-isometric embedding $...
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2 votes
1 answer
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$\delta$ hyperbolic geodesic metric spaces

I have a basic question about the definition of $\delta$ hyperbolic geodesic metric spaces using triangles (studied in geometric group theory cours). The definition I studied in class is that a ...
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1 vote
1 answer
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Proper and cocompact group action by isometries on a metric space

I am reading the book "Metric spaces of Non-Positive Curvature" by Bridson and Haefilger and got stuck with the following : Proposition II.6.10(2) Let $\Gamma$ be a group that acts properly ...
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Group acting properly discontinuous but not cocompactly on a metric space

Let $\Gamma$ act properly discontinuous but not cocompactly on a metric geodesic space $(X,d)$ by isometries s.t every closed ball in $X$ is compact. For every $x_0\in X, r\in\Bbb{R}$, define $D(x_0,r)...
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Quasi-isometry of finitely generated group

Let $\Gamma$ be a finitely generated group, with two generating sets $S_1,S_2$. Deduce, from the Milnor - Svarc lemma, that $Cay(\Gamma, S_1)$ and $Cay(\Gamma,S_2)$ are quasi isometric, where $Cay(\...
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Does $A$ have infinite order in $G= \langle A,B \ |\ B A B^{-1} = A^2 \rangle $?

I have a group (arising from the fundamental group of a manifold) $$G= \langle A,B \ |\ B A B^{-1} = A^2\rangle $$ and I would like to show that $A$ is an element of infinite order inside $G$. ...
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2 votes
1 answer
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Do there exists conditions we can put on two groups which have the same growth rate, so that their Cayley graphs are isomorphic?

Given a finitely generated group $G$ with a generating set $S$, we can define the growth rate function of a group, denote it $\#_{G,S}(n)$. It is clear that two groups having the same growth rate ...
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1 vote
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Reference for solvability of Baumslag-Solitar groups [duplicate]

I've been searching for a while and couldn't find any reference which provides necessary and sufficient conditions over $m,n$ for the group $BS(m,n)$ to be solvable. The only I could find is that it ...
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Torsion free groups with no unique products (notation)

I am reading a paper by William Carter titled "New examples of torsion-free non-unique product groups" and saw the following group: $$P_k=\langle a,b\mid ab^{2^k}a^{-1}b^{2^k},ba^{2}b^{-1}a^{...
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1 vote
1 answer
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Commensurable, Quasi-Isometric and Finitely Generated Groups

Metric Spaces of Non-Positive Curvature, Book by André Haefliger and Martin Bridson, page $141$. Two groups are said to be Commensurable if they contain isomorphic subgroups of finite index. ...
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Does the Baumslag Solitar group $B(2,3)$ contain a non-trivial element with arbitrary roots?

The Baumslag Solitar groups $B(n,m)$ are defined via the presentation $\langle a,b \mid b a^m b^{-1} = a^n \rangle$. We say that an element $g$ of a group $G$ has an $n$-th root, if the equation $g = ...
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2 votes
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Understanding an example of a pencil where addition is that of an elementary abelian group

I'm trying to understand the Latin square in picture 2.1 of the paper "On Projective Planes of Order Nine" by Marshall Hall, Dean Swift, and Raymond Killgrove (https://www.ams.org/journals/...
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A Bijective quasiisometry between finitely generated groups is a biLipschitz equivalence

I want to show that any bijective quasiisometry $f:G\to H$ (whith $G$ and $H$ groups, finitely genered by S and T respectely) is a biLipshitz equivalence. I know this: There are constants $\lambda\geq ...
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The homomorphisms $\Gamma\to\mathbb{R}$ are linear combinations of the functions $f_g$

I am reading Brooks and Series, Bounded Cohomology for Surface Groups. At the end of the first paragraph on the last page, it says "the homomorphisms $\Gamma\to\mathbb{R}$ are linear combinations ...
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