Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
1answer
34 views

Groups containing finite index subgroups isomorphic to $\mathbb{Z}$

There are two propositions I'm trying to prove that seem fairly intuitive to me, but I haven't been able to prove: Let $G = A \ast B$ with $A,B$ infinite. No subgroup of $G$ with finite index is ...
2
votes
0answers
50 views

Dynamical systems from an algebraic perspective

I am interested in learning more about dynamical systems. Most of my background is algebraic, specifically in group theory/geometric group theory. I was wondering if anyone knew of a reference that ...
2
votes
1answer
37 views

Reference Request for Small Cancellation Theory

I am looking for a self contained survey / paper / lecture notes on small cancellation theory and it's generalizations. I am aware of Lyndon and Schupp's textbook chapter and I have been recommended ...
1
vote
1answer
64 views

Why coarse maps have to be proper?

A map $f: X \to Y$ between metric spaces is said to be coarse, if the following two conditions hold: $f$ is bornologous, i.e. $$\forall_{R>0} \; \exists_{S>0} \; d(x,y) < R \Rightarrow d(f(...
4
votes
0answers
89 views

Is a HNN extension of a virtually torsion-free group virtually torsion-free?

Let $G=\langle X\ |\ R\rangle$ be a (finitely presented) virtually torsion-free group. Let $H,K<G$ be isomorphic (finite index) subgroups of $G$ and let $\varphi:H\rightarrow K$ be an isomorphism. ...
0
votes
0answers
51 views

Fundamental results in Geometric Group Theory [closed]

I'd like some advices about Geometric Group Theory / Coarse Geometry. I have to prepar a seminar of 2 hours on "something about Geometric Group Theory". So I have to choose a topic (I know that ...
0
votes
0answers
29 views

Number of ends is a quasi-isometry invariant. [closed]

How do I show that if $G \cong_{\text QI}H$, then $G$ and $H$ have the same number of ends?
0
votes
0answers
27 views

How do I show that the direct sum of F2 with itself is a one-ended group? [closed]

How do I show that $\mathbb{F}_2 \oplus \mathbb{F}_2$ is a one-ended group? I can see why visually but I have no idea how to argue it mathematically.
3
votes
2answers
97 views

Show that $\pi_1(M)=\langle a,b|b^{2}=1 \rangle\cong \Bbb Z*\Bbb Z_2$ is non-abelian.

How to show that $\pi_1(M)=\langle a,b|b^{2}=1 \rangle\cong \Bbb Z*\Bbb Z_2$ is non-abelian by taking any two elements of this group that don't commute? Thanks in advance!
4
votes
2answers
124 views

Topology of a free group

I wonder is there any general properties/ restrictions to the possible topologies of a free group (to make it a topological group ofc). More generally do such restrictions exist for any group written ...
2
votes
0answers
19 views

Finding generators of a group from its action on a topological space

Summary I believe I've written a geometric group theory flavoured proof with a mistake in it, but I'm struggling to see why it might be wrong. I haven't found a counter example, but it also feels too ...
0
votes
0answers
28 views

hyperbolic groups are centerless (reference request)

I am looking for a reference for the statement: Any word hyperbolic group has trivial centre (which seemed to be implicit in some old math overflow answer I am reading). I cannot seem to find it by a ...
2
votes
1answer
51 views

How does convergence of sequences define a topology? Question about the definition of the topology of the Gromov boundary.

I am not sure how the second bullet defines a topology on $\partial X$? What exactly is the topology? What are the open sets, and/or, what are the closed sets? How does convergence of sequences ...
1
vote
1answer
28 views

$\infty$-ended subgroups of one-ended groups

Let $G$ be a one-ended hyperbolic group. Can $G$ contain an $\infty$-ended group? If it can, are there any conditions on $G$ beyond hyperbolic which makes it impossible? As a particular example, if $...
2
votes
1answer
41 views

Small diameter, mixing time, and Expander graphs

Let $\Gamma_n$ be a family of $d$-regular finite simple graphs. 1). $\Gamma_n$ has logarithmic diameter if $diam(\Gamma_n) = O(\log |\Gamma_n|)$; 2). $\Gamma_n$ has logarithmic mixing time if $$ \...
1
vote
0answers
36 views

Why do we use two constants in the definition of quasi-isometric embedding?

A map $f: X \to Y$ is a quasi-isometric embedding if there exists constants $K \ge 1$ and $C\ge 0$ so that for all $x, x' \in X$ we have $$\frac{1}{K}d_X(x,x') -C \le d_Y(f(x), f(x')) \le K d_X(x,x') +...
2
votes
0answers
33 views

Characterize edge-transitive Cayley Graphs

Let $G$ be a finite group and $S \subseteq G$ a symmetric subset. The Cayley graph $\Gamma(G,S)$ is always vertex-transitive, but it sufficient a simple example to show that it is not always edge-...
0
votes
1answer
49 views

Presentation of a group generated by reflections through hyperplane

Question: Let $P_i\in\mathbb{R}^n$ be the hyperplane $x_i - x_{i+1} = 0$. Find a presentation for the group $G$ generated by the reflections in $P_1, \ldots, P_{n-1}$ Attempt: I really don't know how ...
0
votes
0answers
30 views

Exact value of exponential growth rate depends on generating set

I am trying to solve an exercise from Clara Loh's Geometric Group Theory: An introduction. The problem uses the exponential growth rate of a finitely generated group $G$ with generating set $S$. The ...
0
votes
0answers
46 views

H. Abels' paper about Specker compactifications

Is there an English translation for Herbert Abels' paper: Specker-Kompaktifizierungen von lokal kompakten topologischen Gruppen?
0
votes
0answers
54 views

Geometry of Groups for Beginners [duplicate]

I'm a second-year undergrad in Mathematics. I recently attended a lecture on the geometry of groups. Needless to say, I understood very little but was amazed nonetheless. The idea that a structure ...
1
vote
0answers
23 views

If $b, c \in G$ act on $T$ and fix disjoint (but nonempty) sets of vertices, then $bc$ doesn't fix any vertex

Suppose we have a group $G$ acting on a tree $T$ without inversions (i.e. it is never the case that $g \in G$ flips an edge). Let $b, c \in G$ such that $b$ and $c$ individually fix at least 1 vertex, ...
0
votes
0answers
31 views

Help with properly discontinuous action implication in Milnor-Schwarz lemma.

From Office Hours with a GGT I am not seeing how proper discontinuity implies there are finitely many translates of $B$ that have distance at most $D$ from $B$. Proper discontinuity says that ...
2
votes
1answer
28 views

Non nesting action on Real Trees

Background Let $G$ be a group acting by homeomorphisms on an $\mathbf{R}$-tree $T$. The element $ g \in G $ is elliptic if the fixed point set $\operatorname{Fix}(g)$ is non-empty. The group action ...
4
votes
1answer
52 views

Fixed points of a group action on tree

Suppose a group $H$ acts on a tree $T$, and this action fixes a point. Let $T_1$ be an $H$ invariant subtree of $T$. How do I show that $H$ fixes a point in $T_1$?
1
vote
1answer
54 views

Decompose a 1D polynomial into irreducible representations

I am taking an introductory course for group- and representation theory and struggle with this problem: Consider the integral: $\int_{-1}^1 P_2(x)dx$ where $P_2(x)=\alpha + \beta x + \gamma x^2$ ...
2
votes
1answer
64 views

How can I write $\mathbb{Z}^2 \ast \mathbb{Z}^2$ an HNN-extension with infinite cyclic associated subgroups?

I have a free product $\mathbb{Z}^2 \ast \mathbb{Z}^2$, and I need to express it as an HNN-extension with associated subgroups isomorphic to $\mathbb{Z}$. I have not had much success. The way I am ...
3
votes
1answer
60 views

Finite subgroup of a group of homeomorphisms of closed unit disc

Example of a non-trivial finitely generated subgroup of finite order of a group of homeomorphisms (which fix boundary point wise) of closed unit disc? This question is related to this one (Finitely ...
0
votes
2answers
26 views

Infinite, finitely generated subgroup of a group of homeomorphisms of closed unit disc

Can we construct a finitely generated subgroup of infinite order of a group of homeomorphisms (which fix boundary point wise) of closed unit disc?
2
votes
0answers
64 views

Group theoretic finiteness properties stronger than linearity or weaker than Hopficity?

Generally speaking (modulo some overlap), there are two types of finiteness properties for a finitely generated group $G$, homotopical properties (e.g. type $F_n$, type $F$, $cd_\mathbb{Z}<\infty$) ...
0
votes
0answers
52 views

Example of a homeomorphism of a plane which fixes unit circle point wise

What would a non trivial example of a homeomorphism of a plane which fixes unit circle point wise. If I take \begin{equation} h(x,y)=\begin{cases} (x,y)& \text{$(x,y)\in S^1$}\\ (x+2,y+2) & \...
0
votes
1answer
63 views

How to characterize the group of homeomorphisms of unit disk in terms of the group of homeomorphisms of plane?

Let $G$ be the group of homeomorphisms of unit disk $(D)$ fixing boundary point wise and $P$ be the group of homeomorphisms of plane $\mathbb{R}^2$. Can we characterize $G$ in terms of $P$. Speaking ...
0
votes
1answer
59 views

Characterization or examples of metric spaces with this property

Let $X$ be a metric space and let $J$ be a subgroup of $\text{Isom}(X)$. For any $x \in X$ and compact subset $K \subset X$, consider the set $$A = \left\{ g \in J : g(x) \in K \right\}.$$ What ...
2
votes
1answer
77 views

Is there any relationship between growth rate and amenability?

Let $G$ be a finitely generated group, I'm interested in whether there is any relationship between amenability of $G$ (as a discrete group) and its growth rate. To make the question more precise let's ...
2
votes
0answers
71 views

Word length vs hyperbolic length of curves on a hyperbolic surface

Suppose S is a surface which admits a hyperbolic metric - by this I mean a complete Riemannian metric of constant negative curvature, with totally geodesic (possibly empty) boundary. Fix some ...
2
votes
1answer
51 views

Is the infinite regular tree $T_\infty$ quasi-isometric to the $n$-regular tree $T_n$?

It is known that for $n\geq 3$ all $n$-regular trees $T_n$ are quasi-isometric to each other. This can e.g. be seen by using an edge contraction argument. Is there also a quasi-isometry between the ...
4
votes
0answers
80 views

Finite Presentation of a subgroup

I have the group $\langle a,b \mid a^3b^3\rangle$ Now I send both $a$ and $b$ to the generator of $\mathbb{Z}/3\mathbb{Z}$. This gives a well-defined homomorphism from our group to $\mathbb{Z}/3\...
2
votes
0answers
57 views

Newman's “proof” that surface groups are LERF?

In trying to find alternatives to Scott's paper, I came across Tretkoff Covering Spaces, Subgroup Separability, and the Generalized M. Hall Property, which references this paper of Newman to show ...
1
vote
1answer
166 views

Request for applications of the Filling Theorem: geometric and algebraic Dehn functions are asymptotically equivalent

I am studying geometric group theory and metric spaces of bounded curvature. In particular, I was reading the seminal Gromov's Hyperbolic Groups and trying to filling the details. In the first page, ...
3
votes
0answers
92 views

Which 3-manifolds can be cubulated?

I am trying to get a picture of what is currently known about cubulability of 3-manifolds, though cannot seem to find a good overview. I am personally most interested in compact 3-manifolds with ...
3
votes
0answers
105 views

Show that $\mathbb{Z} = \langle a, b \mid a^{12} = b,\ ab = ba \rangle$ has dead end elements

This exercise is taken from the book "Office Hours with a Geometric Group Theorist" (Office Hour 15, exercise 8): Exercise: Show that the group $\mathbb{Z}$ has dead end elements with respect to the ...
1
vote
1answer
118 views

Are 3-sheeted covers of figure 8 homotopy equivalent to each other?

There are several threads about classification of all 3-sheeted covers of figure 8. For example, this one: How to classify 3-sheeted covering space for $S_{1}\vee S_{1}$? It seems to me if tighten a ...
2
votes
1answer
27 views

Implications of a local isomorphism on discrete subgroups and Kazhdan property (T)

In the book "Discrete groups, expanding graphs and invariant measures" by Alexander Lubotzky, page 37, the author says that all finitely generated discrete Kazhdan subgroups of $SO(3)$ are finite, and ...
4
votes
1answer
85 views

Why is this sequence of isometries contained in a compact set?

I'm reading through a proof of the Mostow rigidity theorem (pages 738-740) in https://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf but I'm stuck at a certain part. In Step 2, we consider the vertical ...
1
vote
1answer
28 views

Word norm in compact subsets of finitely generated groups

Let $G$ be a finitely generated topological group, not necessarily discrete. Fix a finite generating set $S$ and denote by $|x|$ the word norm of $x \in G$ with respect to this generating set, i.e., ...
0
votes
0answers
27 views

Quotient of an amenable group (non-discrete case) and Haar measure on the quotient

"Let $G$ be a locally compact amenable group with Haar measure $\mu$, $H$ a closed normal subgroup, then $G/H$ is amenable." I am trying to prove this fact, and there are two definitions I can use (...
0
votes
1answer
49 views

finding the geometric dimension of $\mathbb{Z}^n$

I've just been introduced to the idea of geometric dimension of a group (smallest dimension of $K(G,1)$) and I'm trying to figure out what the geometric dimension of $\mathbb{Z}^n$ is for $n\geq 3$. ...
0
votes
1answer
33 views

Finitely presented groups which are neither Hopfian nor cohopfian

Are there any examples of (preferably countable) finitely presented groups which are neither hopfian nor cohopfian? If so, is there a classification of such groups?
1
vote
1answer
39 views

Every quasiisometry is a quasiisometric embedding

Definition. Let $X,Y$ metric spaces. A map $f:X\to Y$ is called $(L,C)$-coarse Lipschitz if $$d_Y(f(x),f(x'))\leq Ld(x,x')+C$$ for all $x,x'\in X$. A map $f:X\to Y$ is called $(L,C)$-quasiisometric ...
0
votes
1answer
62 views

The kernel of a surjective homomorphism G --> Z is finitely generated

Let $G$ be a finitely generated group. The following is a part 1 of proposition 6.3.10 in Clara Löh's Geometric Group Theory and is left as an exercise, but I can`t figure it out. Let $G$ be a ...