# 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.

750 questions
Filter by
Sorted by
Tagged with
30 views

### $T_4/\langle\{b^nab^{-n}\mid n\in\mathbb{Z}\}\rangle$ and the real line with a loop attached to each integer point

Bowditch uses an example in his A Course on Geometric Group Theory, to explain a fact that a subgroup $G\leq F$ need not be freely generated even if $F$ is, but I cannot understand some details of it. ...
• 420
72 views

### Geodesics in Hyperbolic Disk

Let $\Gamma \le \operatorname{PSL}(2,\mathbb{R})$ be a discrete subgroup and $\Sigma:=\mathbb{D}^2/\Gamma$ be the quotient. Then $\Sigma$ is a hyperbolic surface whose universal cover is $\mathbb{D}^2$...
• 440
117 views

### Torsion elements of $SL_3(\mathbb{F}_p[x])$? (Quick question)

Is every element of $SL_3(\mathbb{F}_p[x])$ a torsion element? Here are my thoughts: First of all, the group is noncommutative, so a torsion element is an element of finite order. I'm thinking of ...
65 views

### Paths in the hyperbolic disk

Let $\Gamma \le \operatorname{PSL}(2,\mathbb{R})$ be a discrete subgroup and $\Sigma:=\mathbb{D}^2/\Gamma$ be the quotient. Then $\Sigma$ is a hyperbolic surface whose universal cover is $\mathbb{D}^2$...
• 440
1 vote
50 views

46 views

### Injective homomorphism between profinite groups

Given an inclusion $P\subset \Gamma$, it induces a homomorphism $\phi:\hat P\to \hat\Gamma$ on profinite completions. Then is that true that $\phi$ is injective iff given any normal subgroup of finite ...
• 1,897
163 views

### Uniform Følner condition and amenability

I am currently studying the various characterizations of amenability, and I am in particular working with Følner sets: a group $G$ is amenable if and only if for every finite subset $S \subseteq G$ ...
1 vote
68 views

### $\Gamma <\mathrm{PSL}_2(\mathbb{R})$: non-compact if contains parabolic element.

It seemingly is a fact that a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$ acting on hyperbolic space cannot be compact if it contains a parabolic element. I was wondering if the following proof ...
• 868
166 views

### A Conjecture in Low-Dimensional Topology.

Context I looked through a book called "Problems in Low-Dimensional Topology," where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
69 views

### Proof of a particular piece of Milnor-Wolf theorem

The Milnor-Wolf theorem says that a finitely generated solvable group that doesn't have exponential growth is virtually nilpotent. The proof I've seen is divided into two pieces: Prove that such a ...
• 899
96 views

### $F_2\ltimes F_2^{2n-4}$ is a subgroup of $\mathrm{Aut}(F_n)$.

In a paper I read that $F_2\ltimes F_2^{2n-4}$ is a subgroup of $\mathrm{Aut}(F_n)$. The proof of this fact is as follows: Choose $F_2\leq \mathrm{Aut}(F_2)$ and let it act diagonally on $F_2^{2n-4}$, ...
• 1,859
34 views

### Geometric centroid and symmetry

I have some intuition that for shapes with some rotational symmetry, the symmetry is typically about the geometric centroid. For example, for a cuboid and cube this is true. What about in general, for ...
• 699
61 views

### Existence of a probability as a condition for amenability of a group

I'm trying to understand the proof of this statement: Let $G$ be a countable group, then: $$G \text{ amenable }\iff \exists \nu \in \text{Prob}(G) \text{ s.t }(G,\nu) \text{ is Liouville}$$ For that ...
• 627
77 views

### Length space with proper and cocompact group action

Given a length space $X$. Suppose there is a group $G$ acting properly and cocompactly on $X$ by isometries, then $X$ is locally compact and complete. How can we prove this statement? Any idea will be ...
29 views

### Constructing Quotient Graph from Tree of Representatives (Theorem 4 proof, page 27, Serre's Trees)

It is claimed in the below proof that if we contract each $gT$ to a single vertex, this forms $X/G$. I can't see why this is true. I know you can form X/G by identifying vertices in $X$ which are in ...
176 views

61 views

### Schutzenberger graphs of an Inverse Semigroup?

I recently came across the idea of extending the well-known Cayley graph construction for semigroups and learned that the outcome does not have all the expected properties even for the nice classes of ...
• 18.3k
1 vote
59 views

### parallel walls in Coxeter groups

I am trying to understand the proof of 3.2 in the paper "Coxeter groups are biautomatic" by Osajda and Prytycki [OP]: https://arxiv.org/abs/2206.07804 I will reformulate the statements with ...
• 11
73 views

### Euler-Poincaré formula for foliations

Does someone have a nice proof for Proposition 11.14 in Farb&Margalits "Primer to Mapping Class Groups", which states the following: Let $S$ be a closed surface with a singular foliation ...
• 1,424
73 views

### Discrete subgroups of Lie groups other than Lattices

I know a lattice is a discrete group of $G$ which is of finite covolume. I am just curious to know examples of discrete subgroups of $SL_2(\mathbb{R})$ or in particular $SL_n(\mathbb{R})$ other than ...
1 vote
63 views

### p-adic Lie groups v.s. real Lie groups in philosophies

What makes $p$-adic Lie groups interesting objects to study? I am not familiar with the representation theory or the number theory side of things, so I'd like to frame my question within the contexts ...
• 139
114 views

### Cocompact action with finite stabilizer implies locally finite

Given a discrete group $G$ acting cellularly and cocompactly on a cell-complex $X$ with finite stabilizers, I am struggling to show $X$ is locally finite. I am trying to show any vertex $x\in X$ is ...
• 599
34 views

### Closest Equivalent to Cayley Graphs for Partial Groupoids?

[A partial groupoid (half-magma) is a set S equipped with a (single-valued) partial binary operation, as in Bruck's Survey of Binary Systems.] This question may be nonsensical, given that the duality ...
• 21
67 views

### Finite index subgroups of amalgamated free products over finite index subgroups

Let $G = H_1 \ast_K H_2$ be an amalgamated free product of two groups such that $K$ has finite index in both $H_1$ and $H_2$. Let $G'$ be a finite index subgroup of $G$. Does it follow that $G'$ ...
• 193
1 vote
56 views

### Dehn presentation implies finitely many conjugacy classes of elements of finite order

Let $G$ be a finitely generated hyperbolic group. Show that $G$ contains only finitely many conjugacy classes of elements of finite order. In “Geometric Group Theory: An Introduction” by Clara Löh, it ...
• 613
1 vote
63 views

### Trying to find the set of unique representatives for the geodesics in the group $\langle a,t \mid ata^{-2}t^2a^{-2}tat^{-4}\rangle$

I am studying the conjugacy growth of the groups, and I encountered the following group: $$G=\langle a,t \mid ata^{-2}t^2a^{-2}tat^{-4}\rangle$$ Thanks to Derek for pointing out that $G$ is an ...
• 1,431
49 views

### Maps from a finitely generated pro-p group to $\mathbb F_p$ factors through Frattini quotient

Let $G$ be a finitely generated pro-p group, these notes (p.99, Corollary 5.4.21) claims that all maps from $G$ to $\mathbb F_p$ factor through the Frattini quotient ($G/\Phi(G)$), where $\Phi(G)$ is ...
• 599
81 views

### How to visualize the 6 roto-reflections in the group of symmetries of a tetrahedron $S_4$?

I'm working on an applet that will calculate the product of two symmetries. (It's unfinished but here's a link to the project if you're curious.) I want the applet to show visuals to help the user ...
125 views

### Hyperbolic surface subgroups of products of groups

I am an undergraduate student, currently working in an REU project about geometric group theory. I know a few basic notions of geometric group theory and algebraic topology: Cayley graphs, fundamental ...
• 130
58 views

### References for theory behind Geometric Deep Learning

I'm currently reading a blog post on "Geometric Deep Learning," which I find fascinating yet challenging to comprehend without a solid mathematical background. I am halfway through my ...
• 627
44 views

### Subgroup of PSL(2,R) generated by two elements with elliptic commutator is not free.

Suppose $[A,B]$ is elliptic for some $A,B \in PSL(2;\mathbb{R})$. Then $\langle A,B \rangle < PSL(2;\mathbb{R})$ is not free. Is there any nice way to see this? If $[A,B]$ is elliptic of finite ...
• 279
1 vote
117 views

### Is the isomorphism problem solvable for Euclidean groups?

Suppose you had two group presentations, and you know they are Euclidean groups, can you tell if they are isomorphic or not? It has been suggested to me that it is probably possible to tell if they ...
• 613
172 views

### What is a simplicial graph?

I keep seeing the term "simplicial graph" thrown around in the context of geometric group theory. I assume this means a graph which is also a simplicial complex (a graph with no loops), but ...
• 948
76 views

### Epimorphism between free groups that inject on a finite subset

I asked a question on MathOverflow (https://mathoverflow.net/q/454012/513011) where the following lemma appeared: Folklore lemma: Let $S$ be a finite subset of the free group $F_n$ of finite rank $n$....
1 vote
64 views

• 53
1 vote