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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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$T_4/\langle\{b^nab^{-n}\mid n\in\mathbb{Z}\}\rangle$ 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. ...
一団和気's user avatar
3 votes
0 answers
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 $\mathbb{D}^2$...
Ramanasa's user avatar
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5 votes
2 answers
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 ...
June in Juneau's user avatar
2 votes
1 answer
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 $\mathbb{D}^2$...
Ramanasa's user avatar
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0 answers
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Relations in the fundamental group of a surface

Let $\Sigma$ be a smooth oriented surface (possibly with boundary) and $G:=\pi_1(\Sigma)$ its fundamental group. Suppose that two nonzero elements $\alpha$, $\beta\in G$ satisfies $\alpha \beta = \...
Ramanasa's user avatar
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1 vote
0 answers
54 views

Can someone explain these Definitions about Cell Complexes for me?

I've been reading "On a small cancellation theorem of Gromov" by Yann Ollivier (https://projecteuclid.org/journalArticle/Download?urlId=10.36045%2Fbbms%2F1148059334 ) recently and these ...
shekh's user avatar
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0 answers
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Prove the length of LCS and UCS are equal

In my lecture notes on growth of groups, we are given two definitions for a $c$-step nilpotent group $G$. It goes as follows: A group $G$ is called $c$-step nilpotent if its lower central series, $\...
soggycornflakes's user avatar
2 votes
0 answers
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 ...
Danny's user avatar
  • 1,897
3 votes
0 answers
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 $S \subseteq G$ ...
Alice in Wonderland's user avatar
1 vote
1 answer
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$\Gamma <\mathrm{PSL}_2(\mathbb{R})$: 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 ...
Pastudent's user avatar
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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 ...
Amanuel Jissa's user avatar
2 votes
1 answer
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 ...
Hempelicious's user avatar
2 votes
0 answers
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$F_2\ltimes F_2^{2n-4}$ 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 $F_2^{2n-4}$, ...
Marcos's user avatar
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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 ...
user3180's user avatar
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3 votes
1 answer
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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 ...
bml64's user avatar
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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 ...
Patrick Star's user avatar
0 votes
1 answer
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 ...
Integer Indexed's user avatar
2 votes
1 answer
176 views

Explanation of these Mayer-Vietoris Maps

I am reading about the Mayer Vietoris sequence $$\cdots \xrightarrow{\beta}H_c^{q-1}(A\cap B) \xrightarrow{\delta} H^{q}(A\cup B) \xrightarrow{\alpha}H^{q}(A) \oplus H^{q}_c(B) \xrightarrow{\beta}H^{q}...
June in Juneau's user avatar
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Checking the image of mapping class in $\text{Aut}(F_{2g})$ stabilizes boundary curve

Overview: the mapping class group maps into $\text{Aut}(F_{2g})$ and its image stabilizes the surface relation. I am trying to check this for a specific example and am doing something wrong. The ...
Chase's user avatar
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0 answers
104 views

Inequalities regarding the geometric dimension of groups

I'm studying some group cohomology, and I'm stuck on the following problem (namely, problem VIII.1.3 of Brown's Cohomology of Groups): let's define the geometric dimension of a group $G$ as $$ \text{...
Wallace's user avatar
2 votes
1 answer
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 ...
Bumblebee's user avatar
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1 vote
0 answers
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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 ...
Tobi's user avatar
  • 11
3 votes
0 answers
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 ...
ctst's user avatar
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2 votes
0 answers
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 ...
User 11111's user avatar
1 vote
0 answers
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 ...
duang's user avatar
  • 139
3 votes
2 answers
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 ...
Kat's user avatar
  • 599
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0 answers
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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 ...
shea's user avatar
  • 21
2 votes
1 answer
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 $G'$ ...
Jean Charles's user avatar
1 vote
0 answers
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 ...
cede's user avatar
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1 vote
0 answers
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 ...
ghc1997's user avatar
  • 1,431
3 votes
1 answer
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 ...
Kat's user avatar
  • 599
6 votes
1 answer
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 ...
Evyenia Coufos's user avatar
0 votes
1 answer
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 ...
Mergolyx's user avatar
  • 130
0 votes
0 answers
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 ...
bayes2021's user avatar
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0 votes
0 answers
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 ...
Math's user avatar
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1 vote
0 answers
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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 ...
cede's user avatar
  • 613
0 votes
0 answers
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 ...
Mithrandir's user avatar
5 votes
1 answer
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 $n$....
TheMathematician's user avatar
1 vote
0 answers
64 views

Help to understand the geodesics in $BS(1, 2)$

I would like to understand the sets of geodesics in $BS(1, 2)$, which is described in https://arxiv.org/pdf/1908.05321.pdf, Proposition 3 (page 3). Write $$ G=B S(1, 2)=\left\langle a, t \mid t a t^{...
ghc1997's user avatar
  • 1,431
0 votes
0 answers
86 views

Alexander Polynomial

Recently I learned the Alexander Polynomial of a knot and how to find the polynomial for a given knot. Now there are some questions arise, I am trying to give some classification of hyperbolic knots, ...
T ghosh's user avatar
  • 51
0 votes
0 answers
72 views

linear isoperimetric inequality implies hyperbolicity

I am trying to find a nice proof that a finitely presented group satisfying a linear isoperimetric inequality implies it is hyperbolic. I came across these lecture notes, Theorem 3.22, but I am having ...
cede's user avatar
  • 613
3 votes
1 answer
86 views

Large-scale Lipschitz and bornologous maps

I'm trying to understand the following questions which I took from Roe's book "Lectures on coarse geometry". A map of metric spaces $f:X \to Y$ is called large-scale Lipschitz if there are ...
Rise23's user avatar
  • 93
3 votes
1 answer
89 views

Virtually $\mathbb{Z}^2$, $\mathbb{Z}^3$ using Bass-Guivarch Formula

My question is about the Bass-Guivarch formula which relates the growth $d(G)$ of a polynomial growth group $G$ to the torsion-free rank of consecutive quotients in its lower central series: \begin{...
Owen Huang's user avatar
2 votes
0 answers
121 views

Cayley graph of a finitely generated group $G$

Good time of day! I'm trying to show that: "The Cayley graph of a finitely generated group $G$ is quasi-isometric to a line if and only if $G$ has a cyclic subgroup of finite index". My ...
Victory's user avatar
  • 411
1 vote
0 answers
47 views

Quasi-geodesic rays are closed to geodesic rays in proper hyperbolic geodesic spaces

We define the boundary of a hyperbolic metric space $\partial X$ as the equivalence classes of geodesic rays up to finite Hausdorff distance and $\partial_q X$ as the equivalence classes of quasi-...
quuuuuin's user avatar
  • 637
4 votes
0 answers
44 views

Strong converse of Kazhdan's property (T)

In his 1972 paper Sur la cohomologie des groupes topologiques II, Guichardet proved$^\ast$ that free groups satisfy the following strong converse of property (T): The $1$-cohomology $H^1(\mathbb F_d,\...
MaoWao's user avatar
  • 15.2k
8 votes
1 answer
150 views

Does every sequence of group epimorphisms (between finitely generated groups) contain a stable subsequence?

I have a question that is related to the topic of limit groups: Let $G$ and $H$ be finitely generated groups and let $(\varphi_n: G \to H)_{n \in \mathbb{N}}$ be a sequence of group epimorphisms. Does ...
TheMathematician's user avatar
3 votes
1 answer
67 views

Double of a group vs twisted double

Let $A$ be a group with a subgroup $B$, we define the double of $A$ along $B$ as the free amalgamated product of two copies of $A$ along $B$, i.e., if $$ A=\langle S \mid R\rangle $$ and we use ...
Dubois's user avatar
  • 53
1 vote
1 answer
93 views

$\delta$-hyperbolic group is finitely presented

The following corollary is from Discrete groups by Ohshika. Corollary 2.70. Hyperbolic groups are finitely presented. The author didn't prove it but said that 'Combining this theorem with the ...
one potato two potato's user avatar
2 votes
0 answers
57 views

Can the growth of a group be bounded concretely when the group is of subexponential growth?

When talking about the growth of a group, the cases of exponential and polynomial growth are described by concrete bounds on the growth function, however, this is not the case for intermediate growth (...
floresllarena's user avatar

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