# 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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94 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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### 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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### 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 ...
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### 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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### Applications of Tits' alternative in number theory

I have recently studying Tits' alternative. The theorem statement goes like the following: Tits' alternative: Let $G$ be any finitely generated linear group over a field. Then one of the following is ...
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### Hyperbolic structure on surface gives a complex structure

My question is from A primer on mapping class group, p.295: I can see $X=\Delta/\Gamma$ has an induced hyperbolic structure, but why conversely any such hyperbolic structure gives a complex structure ...
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### Groups with two ends: showing that either $E\Delta gE$ is finite or $(E\Delta gE)^\complement$ is finite.

Let $G$ be a finitely generated group with $e(G) = 2$, and let $\Gamma$ be a Cayley graph of $G$. There is then a finite subgraph $C$ such that $\Gamma \setminus C$ has exactly two connected, ...
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### Geometric Group Theory, Meier lemma 11.30 about a two-ended groups $G$

This is probably a basic algebraic (or even set-theoretic) matter. I am reading "Groups, Graphs and Trees" by J. Meier. It's about Lemma 11.30 which is left as an exercise to the reader. ...
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### Conjugation Classes inside the Orbit of close curve in Hyperbolic Surface under $Mod(S)$

I've heard the statement: Let $S$ be a hyperbolic surface with boundary and $\left[ \alpha \right] \in \pi_1 \left( S \right)$ a non-trivial element that is not conjecture to boundary element, then ...
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### Are all intermediate growth branch groups just-infinite?

Are all branch groups of intermediate growth just-infinite? I can't seem to find an answer to this one way or another; the question is motivated by the fact all examples of intermediate growth branch ...
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### Understanding the proof of solvable conjugacy problem for hyperbolic groups.

https://www.math.ucdavis.edu/~kapovich/280-2009/hyplectures_papasoglu.pdf https://courses.maths.ox.ac.uk/node/view_material/48431 In the first link the theorem I am talking about is on page $29$, ...
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### Amalgamated product question.

Let $\alpha: C \rightarrow A$ and $\beta: C \rightarrow B$ be monomorphisms and let $G$ be the corresponding amalgamated product of $A,B$ over $C$. Let $A_1$ be a subgroup of index $2$ of $A$, such ...
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### Extending a quasi-isometry of a neutered hyperbolic space

Suppose $\phi : B \to B$ is a quasi-isometry of a neutered space $B$ (so $B$ is obtained by removing a collection of disjoint open horoballs from $\mathbb{H}^n$, and the metric $d_B$ on $B$ is the ...
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### Boundary of a CAT(0) space

Suppose $X$ is a complete CAT(0) space.Then is it true that the cone-topology as, it is defined in Bridson-Haefliger's book metrizable? Also what can we say about Hausdorffness? What if we take a ...
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### Geodesic in CAT(0) space

Suppose $X$ is a CAT(0) space and $\gamma$ be a geodesic joining two point say $x$ and $y$ in $X$.Then, can we find a small ball around $y$ such that geodesic joining the point $x$ and a point of ball ...
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### Group associated to Isometries of Space. Reverse direction.

I have seen several posts seeking to find groups of isometries associated to several metric spaces or figures/subspaces embedded in them. I am trying to go in the opposite direction and, given a group ...
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### Scaling of CAT(0) metric

Suppose $(X,d)$ is a geodesic metric space which satisfies CAT(0) inequality.Now if we rescale the metric $d$ then will it still be a CAT(0) metric space i.e, all geodesic triangles satisfies CAT(0) ...
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### Does the Unique Product Property mean that there are only two elements in the set that can create any other in the set, given an operation?

I was under the impression that the way it's defined in the title was the correct way to interpret the UPP(as that was what I was searching for) from the definition here: https://mathoverflow.net/...
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### First uses of the Ping-Pong Lemma

I am interested in knowing the origins of this useful result, but I haven't been able to precisely pinpoint the context of its first use. Most texts seem to indicate the result originally comes from ...
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### Group of direct symmetries of a regular tetrahedron

I'm trying to understand the group of direct symmetries of the regular tetrahedron ($\mathcal{T}$), $$G_+ := \operatorname{Sym}(G) \cap \mathcal{M}_+,$$ where $\mathcal{M}_+$ is the set of rotations ...
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### Definition of property RD

In the following paper https://www.jstor.org/stable/2001458?seq=1#metadata_info_tab_contents Jolissaint introduces the property RD of a group G if the space $H_L^\infty(G)$ is contained in the reduced ...
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### How can the following lemma be used to solve the conjugacy problem for hyperbolic groups?

We are given the following lemma: Let $G = \langle X \ | \ R\rangle$ be a $\delta$-hyperbolic group, then let $u,v \in X^\ast$ be two words such no shorter words in $X^\ast$ define the same elements,...
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### Arzelà-Ascoli for hyperbolic spaces with natural boundary

I am investigating Paulin's method of certain limits of actions on hyperbolic spaces being (in some sense) actions on $\mathbb{R}$-trees. Let $G$ be a finitely generated group. Part of the proof is ...
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### Colorings versus geometric group theory method

Consider the following famous problem: Two opposite corner cells of a chessboard are removed. Prove that it is impossible to cover remaining cells with dominoes $2\times1$. The standard proof uses ...
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### Finite subgroups of $SL_2({\mathbb R})\times SL_2({{\mathbb R}})$

Is there a classification of finite subgroups of $SL_2({\mathbb R})\times SL_2({{\mathbb R}})$? For instance we have all cyclic groups and all direct products of two cyclic groups. Are there any ...
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### a long product of elements in hyperbolic group is not a proper power

Let $G$ be a hyperbolic group, i.e., there exist $\delta>0$ and a finite generating set $S$ of $G$ such that the Cayley graph $X$ of $G$ relative to $S$ is a $\delta$-hyperbolic space. Assume also ...
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### Examples of groups which are virtually isomorphic but not commensurable

Let $G_1, G_2$ be groups. We say $G_1$ and $G_2$ are commensurable if there exist finite index subgroups $H_1 \leq G_1$, $H_2 \leq G_2$ such that $H_1 \simeq H_2$. We say $G_1$ and $G_2$ are ...
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### Is this the Cayley graph of the braid group on three strands?

I have been attempting to draw the Cayley graph of the braid group $$B_3 = \langle a, b \mid aba=bab \rangle$$ and I obtained something that almost seems too good to be true; here is a picture. This ...
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### On the growth rate of groups

Let $G$ be a countable group that is finitely generated and let $S = \{s_1, \dots, s_d\}$ be a generating set. Suppose also that $S$ is closed under inverses. Consider now $\Gamma(G,S) = (V, E)$ the ...
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### $PSL(2, \Bbb{Z})\cong \Bbb{Z}_2 \ast \Bbb{Z}_3$

I am trying to show that $PSL(2, \Bbb{Z})\cong \Bbb{Z}_2 \ast \Bbb{Z}_3$. What I've shown is that, for $$A= \left(\begin{matrix} 0 & 1 \\ -1 & 0 \\ \end{matrix}\right)$$ ...
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### Fixed points of elliptic isometries in Gromov boundary

Suppose $X$ is a proper, geodesic, $\delta$-hyperbolic, metric space. Let $\partial X$ denote the Gromov boundary of $X$. It is known that isometries of $X$ are exactly one of three kinds: ...
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### Finite index subgroup is quasi-isometric to original group

I have been attempting the following problem for a while: Let $G$ be a group generated by some finite set $X$. If $H \underset{f.i.}{\leq} G$ is a subgroup of finite index, then $H$ is quasi-...
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### Are all groups with relators whose exponents sum to $0$ totally orderable?

I apologize for the lengthy preamble, I have no idea how much of what I am relying on is common knowledge (or even correct to be honest). Let $G=\langle S\mid R\rangle$ be a finitely presented group ...
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### Calculating $\chi(\mathbb{Z} * \mathbb{Z}/m\mathbb{Z})$, the Euler characteristic?

Question 1: For a discrete group $G$, does $\chi(G)$ denote the Euler characteristic of $G$, or is this notation commonly used for something else? For the next three questions, assume $\chi$ does ...
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### Type of an isometry of a $\delta$-hyperbolic space

Let $(X,d)$ be proper geodesic $\delta$-hyperbolic metric space. Let $\gamma \in Isom(X)$. Denote by $\partial X$ the boundary at infinity of $X$ (which is invariant of base-point). Let $x\in X$. We ...
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### Cube complex VS polygonal complex

I know it is very hard to decide, given a metric cell complex, if it is nonpositively curved or not. And by the Notes of Sisto, there is a theorem saying: A compact polygonal complex is CAT($0$) if ...
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### CAT($0$) cube complex and homeomorphism

If $X$ is a CAT($0$) cube complex and $Y$ is homeomorphic to $X$, I think homeomorphism preserves the cube complex structure, simply-connected and the link condition. So can I say $Y$ is also a CAT($0$...
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### Linear representation of special euclidean group as subgroup of GL2(C)

I came across this question: Find an isomorphism from the group of orientation preserving isometries of the plane to some subgroup of $GL_{2}(\mathbb C)$. I'm having trouble with finding such ...
I'm looking for a non-elementary hyperbolic group which is quasi isometric to $\mathbb{H}^2$ (and if possible one quasi-isometric to $\mathbb{H}^3$). I know the group $\text{PSL}(\mathbb{R})$ acts by ...
I've recently started studying Geometric Group Theory, and came across this interesting application of the Švarc-Milnor lemma to Riemannian Geometry. Theorem) Let $M$ be a compact connected Riemann ...