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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1answer
67 views

Quasi-isometric Classification of Free Products of Surface Groups

Let $S_g$ denote the compact surface with $g$ holes, and denote its fundamental group as \begin{equation} \Sigma_g=\pi_1(S_g)=\left<a_1,b_1...,a_g,b_g\Big| \prod_{ i \in \{1,...,g\}} [a_i,b_i]\...
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1answer
36 views

Ends of a group and Spherical growth function

I am trying to get some intuition about a spherical growth function of a group, using the notation from A. Mann $s_G(m)=\sum_{n=0}^m a_G(n)$, where $s_G(m)$ is the cumulative growth function and $a_G(...
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1answer
42 views

When does a RAAG act on a Tree?

Show that a RAAG acts on a tree of valence 4, acting transitively on vertices, if at least one pair of vertices is not joined by an edge. I'm trying first to prove that every such RAAG $G$ acts on a ...
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1answer
59 views

Reflection groups, geometric group theory: John Meier

This image is from John Meier's Introduction to the geometry of infinite groups on p. 47, where reflection groups are being introduced. Now the claim is, that every edge in the tiling can be labelled ...
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1answer
66 views

What is the zeta function for a finite graph?

I am getting myself familiar with the background of the non-backtracking operator on a finite graph. The zeta functions of a finite graph is relevant, though not directly related to my project. ...
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2answers
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Characterization of quasi-isometric subsets.

Suppose $(X,d)$ is a metric space and $Y \subset X$ is quasi-isometric to $X$. That is, suppose \begin{equation} X \sim_{QI} Y \subset X. \end{equation} I was wondering, is the inclusion $i: Y \to X$ ...
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1answer
68 views

Geodesics starting at the same point in $\delta$-hyperbolic space are uniformly $2\delta+D$ close

I am reading the book by Clara Löh on Geometric Group theory and stumbled on the following exercise: Let $\delta,D\in\mathbb{R}_{\geq 0}$ and let $(X,d)$ be a $\delta$-hyperbolic space. Let $\gamma:[...
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1answer
48 views

Examples of non-free group actions on trees with finite edge-stabilizers

I am interested in finding examples of finitely-generated non-free groups $H$ such that $H$ is a finite index subgroup of some group $G$ and $H$ acts without edge-inversion on some tree $T$ with ...
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4answers
116 views

An easy example of a non-quasiconvex subgroup

Let $G$ be a finitely generated group, and consider the surjection $\mu:F(A)\to G$ induced by the set of generators $A$, where $F(A)$ is the free group on $A$. A word $w$ is said to be ($\mu$-)...
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1answer
56 views

Finite generation of vertex groups of a cyclic splitting of a hyperbolic group and generalisations of Grushko Theorem

Let $G$ be a finitely generated word hyperbolic group. Suppose $G$ acts non-trivially (without a global fixed point) on a tree without inversions and with cyclic edge stabilizers. Is it true that the ...
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1answer
99 views

Seeking an example of a group with finite presentation for which the Word Problem is not solvable

In the book Geometric Group Theory of Clara Loh, it is proven that the Word Problem is solvable for hyperbolic groups. It is also stated that the Word Problem is not solvable in all finitely presented ...
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1answer
90 views

Finitely generated group with subexponential growth and surjection onto $\mathbb{Z}$ has finitely generated kernel.

I am trying to solve exercise 6.E.20 from the book of Clara Löh on Geometric Group Theory. Let $G$ be a finitely generated group with subexponential growth that admits a surjective homomorphism $\pi:...
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1answer
35 views

Compact subset of a t.d.l.c. group contained in compact-open subgroup?

Let $G$ be a totally disconnected, locally compact group which is the union of its compact-open subgroups (or perhaps only $\sigma$-compact), and let $C \subset G$ be a compact set. Can one conclude ...
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1answer
65 views

What is an invariant mean on $L^\infty(\mathbb{R})$?

Consider the von Neumann algebra $M:=L^\infty(\mathbb{R})$, which consists of (classes) of essentially bounded measurable functions $\mathbb{R}\to \mathbb{C}$. Here $\mathbb{R}$ has the classical ...
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2answers
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Is every fundamental group of a mapping torus of a surface homeomorphism a CAT(0) group?

TLDR; Is every fundamental group of a mapping torus of a surface homeomorphism a CAT(0) group? Let $S$ be a compact surface of negative Euler characteristic and let $f :S\to S$ be a homeomorphism. ...
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1answer
42 views

Prove the existence of $\Psi\in(\ell_\infty(G))^\ast$ satisfying the amenability conditions

Let $G$ be a finitely generated group satisfying the Folner condition and let $S\subset G$ be a finite generating set of $G$. Denote by $\rho=\{\rho_g\}_{g\in G}$ the right translation action of $G$ ...
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1answer
64 views

Prove that $\phi$ is positive definite. [closed]

Let $\phi: \mathbb{Z}\to\mathbb{C}$ be a finitely supported function. Define $\hat{\phi}:\mathbb{R}/2\pi\mathbb{Z}\to\mathbb{C}$ by the inverse Fourier transform $$\hat{\phi}(\theta)=\sum_{n\in\mathbb{...
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1answer
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For discrete group $G$ and $H\leq G$. Show that $G$ also satisfies the Folner condition if $H$ satisfies it and $[G:H]<\infty$. [closed]

A finitely generated group $G=\langle S \rangle$ is said to have the Folner condition if $\forall \varepsilon>0$, there exists a finite subset $F\subset G$ such that $$\#((S\cup S^{-1})F\setminus F)...
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1answer
38 views

Extension and property FH (fixed point for actions on Hilbert spaces)

Remind that a group $G$ has property FH if any $G$ action on a real Hilbert space has a fixed point. For $\sigma$-compact, locally compact groups this is equivalent to the celebrated Kazhdan's ...
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0answers
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Random elements of (infinite) finitely generated groups

I have been studying connections between Geometric Group Theory and Probability, and I was wondering: what work is there about random elements of finitely generated groups? More formally, let $G=\...
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3answers
190 views

Interesting Theorems on Finitely Generated Abelian Groups?

First time teaching algebraic topology, probably gonna be related to most of my questions on here for a while. I was wondering if anyone knows of particularly interesting theorems or examples from the ...
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1answer
75 views

Is $\mathbb{Z}$ Gromov-Hyperbolic?

If $G$ is a finitely-generated group, then we say that $G$ is Gromov-Hyperbolic if it's Cayley Graph, $\operatorname{Cay}(G, S)$, is a Gromov-Hyperbolic metric space. Now in the case of the group of ...
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1answer
49 views

Group acting on trees with relations $abc=1$ and $cba=1$

Suppose $G$ is a finitely presented group that acts on a tree $T$ by isometries, and let $a,b,c\in G$ with relations $abc=1$ and $cba=1$. If two of $a,b,c$ are hyperbolic, does this imply the third ...
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2answers
67 views

What general techniques are there to show that two finitely-generated groups are bi-Lipschitz equivalent?

We define bi-Lipschitz equivalency as follows. We say two metric spaces $(X,d)$, $(Y,e)$ are bi-Lipschitz equivalent if there exists a bijection $f : X \to Y$ and some $ L \geq 1$ such that $$ \...
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0answers
93 views

If there is a convergence action on the space $X = R^{n+1}\cup S^{n}$, is it true that X is a closed ball?

Let G be a group that acts on a Hausdorff compact space X with the convergence property such that the limit set Y is homeomorphic to the sphere $S^{n}$ and X-Y is homeomorphic to $\mathbb{R}^{n+1}$. ...
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0answers
33 views

Show that F2 has an infinite index free subgroup of rank 2 [duplicate]

I am trying to show that F2 (the free group on 2 elements) has an infinite index, free subgroup of rank 2. As it has to be a free subgroup of rank 2 I'm guessing I need to find 2 elements in F2 that ...
2
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1answer
65 views

Proper action in “Metric spaces of non-positive curvature”

(This question is related to Proper action and compactness) In (I.8.2) of "Metric spaces of non-positive curvature", properly action is defined as: for each $x$, there exists a $r$ such that ...
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1answer
79 views

Show that this quasi-geodesic ray is not Gromov hyperbolic?

Consider the spiral (t, log(1+t)) (given in polar coordinates); it inherits the Euclidean metric from the plane. I have to show that this spiral (a quasi-geodesic) is not Gromov hyperbolic. In other ...
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1answer
85 views

Do quasi-isometries of Gromov hyperbolic groups coarsely preserve the group operation?

Let $G$ be a finitely generated group and $H$ a hyperbolic group in the sense of Gromov, with a fixed word metric $d$. Let $\varphi : G\to H$ be a quasi-isometry. Does there exist uniform constant $C&...
3
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1answer
45 views

Does the group growth rate limit the number of edges going out of a vertex in its Cayley graph?

The growth rate of a group $B_n(G, T)$ is based on the number of vertices that can be reached from a given one by $n$ steps along an edge in the Cayley graph of the group, where $G$ is the group (or ...
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1answer
53 views

malnormal subgroup of amalgamated free product

Consider the amalgamated free product $\Gamma = K\ast_{H\simeq H'} L$. Let $A$ be a malnormal subgroup of K i.e, for all $k\in K\setminus A$, $k^{-1}Ak \cap A ={1}$. Is A malnormal in $\Gamma$? I was ...
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1answer
28 views

Bounded Geodesics in a Quadrangle of Fixed Length.

The following question is a slightly weaker version of the question presented here: Geodesic Quadrangle in a Hyperbolic Space. However, this bound suffices for the statement mentioned in the original ...
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1answer
108 views

What finitely-generated amenable groups arise as subgroups of compact Lie groups?

I am looking for examples of (edit: amenable) finitely-generated subgroups of any compact Lie group which are infinite and not virtually abelian. An example with polynomial growth would be ...
4
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1answer
91 views

Is there a known example of a finitely presented group with subexponential growth that isn't polynomial?

The Grigorchuk group is finitely generated and has subexponential non-polynomial growth but I'm not aware of a finite presentation. Does a finite presentation imply that the group is polynomial or ...
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1answer
44 views

Group acting properly and cocompactly by isometries on a metric space has finitely many conjugacy classes of point stabilizers

Let $\Gamma$ be a group acting properly and cocompactly by isometries on a metric space $X$. Then, $\Gamma$ has finitely many conjugacy classes of point stabilizers. The proof of this well-known ...
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0answers
53 views

When are free abelian normal subgroups “virtually” central?

See my previous question for the motivation for this question: Are infinite cyclic normal subgroups "virtually" central? Let $G$ be a (finitely generated) group and let $K \lhd G$ be a ...
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0answers
38 views

Are infinite cyclic normal subgroups “virtually” central?

Let $G$ be a finitely generated group and let $C$ be an infinite cyclic normal subgroup of $G$. My question is: Does $G$ necessarily have a finite-index subgroup $H$ such that $C \le Z(H)$? If not, ...
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2answers
61 views

Hyperbolic boundaries of infinitely generated groups

I was wondering the following: in every book and paper that I looked into the definition of word hyperbolic groups (in the sense of Gromov) contains the condition that the group is finitely generated. ...
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1answer
64 views

Presentation for a hyperbolic group with 2-sphere boundary.

I am looking for examples of hyperbolic groups that have boundary homeomorphic to the 2-sphere, $S^2$. I would like an explicit presentation of such a group so that I can draw its Cayley graph and ...
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0answers
24 views

“Poincaré disk model of fundamental domain triangles” software… [duplicate]

Is there a software to generate the Poincaré disk model of fundamental domain triangles? $\hskip1.7in$ Specially including "general triangles $(p, q, r)$"...
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1answer
135 views

Does $(xzy)^{s/2}$ preserve some special kind of property like orientation?

Let's look at the following presentation: $$ \Delta^*(p,q,r;s/2)=\langle a,b,c\mid a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=((abc)^2)^{s/2}=1\rangle $$ This is a presentation of a special triangle group $\...
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0answers
67 views

Fundamental group of bouquet of 2 circles not in common point

I tried to calculate a fundamental group of bouquet of 2 circles not in common point. Let call this point a (see picture). It is different from fundamental group in b or I did some mistake? (...
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1answer
78 views

Non-finitely generated relatively hyperbolic groups

As we know that free product $G_1\ast G_2$ is relatively hyperbolic with respect to $\{G_1,G_2\}$. So for non-finitely generated relatively hyperbolic groups we can take this example. But i am ...
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1answer
95 views

Second cohomology of a torsion-free hyperbolic group

Does any body know examples of torsion-free hyperbolic group $G$ such that $H^2(G,\mathbb{R})=0$ (trivial G-action on $\mathbb{R}$)? In fact I am interested in if there are known examples of even-...
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1answer
65 views

one-relator groups which are free-by-cyclic

I am reading an article of Baumslag: Baumslag, Gilbert, "Finitely generated cyclic extensions of free groups are residually finite." Bull. Austral. Math. Soc. 5 (1971), 87–94. and he ...
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1answer
123 views

$\delta$ thin trianges implies solvable conjugacy problem for hyperbolic groups: Confusion about $\delta$-rectangles

I am trying to understand the proof that a linear dehn function implies solvable conjugacy. I am referring to Notes on solvable and automatic groups by Michael Batty, after Panagiotis Papasoglu. Here ...
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1answer
177 views

Is the braid group hyperbolic?

The braid groups satisfy a number of properties that one would expect of a hyperbolic group, liking having a solvable word problem, and having exponential growth. Are the braid groups hyperbolic ...
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1answer
53 views

Embedding of the 1-skeleton of a Coxeter group into its Davis complex

Let $(W,S)$ be a Coxeter system and let $\Sigma$ be the corresponding Davis complex. It is well-known that the Davis complex may be equipped with a piecewise Euclidean metric so that it is a proper, ...
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1answer
140 views

Examples of hyperbolic groups that have boundary homeomorphic to $S^2$?

I am working on understanding Cannon's Conjecture which is the following: Suppose that $G$ is an infinite, finitely presented group whose Cayley graph is Gromov-hyperbolic and whose space at infinity ...
4
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2answers
75 views

Let $G$ be a group with a free subgroup of rank $2$. Let $H\leq G$ be such that $[G:H]<\infty$. Then $H$ also contains a free subgroup of rank $2$.

I am having difficulties in solving the following problem. Let $G$ be a group with a free subgroup of rank $2$. Let $H\leq G$ be such that $[G:H]<\infty$. Then $H$ also contains a free subgroup of ...

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