Questions tagged [geometric-group-theory]
Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations
64 questions from the last 365 days
7
votes
2
answers
539
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The ten most fundamental topics in geometric group theory
What are the ten most fundamental topics in geometric group theory?
This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
7
votes
1
answer
213
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Preserving non-conjugacy of loxodromic isometries in a Dehn filling
Suppose that $g$ and $h$ are non-conjugate loxodromic isometries in a cusped hyperbolic $3$-manifold $M$ of finite volume. Fix a cusp $T$ of $M$. Can I choose a hyperbolic Dehn filling of $M$ along $...
0
votes
0
answers
79
views
Visual boundary vs Bowditch boundary [closed]
Is there any difference between the visual boundary of a relatively hyperbolic group and the Bowditch boundary of a relatively hyperbolic group?
Visual boundary is generally associated to a CAT(0) ...
- 1
3
votes
1
answer
269
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$\mathrm{SL}(2,\mathbb{Z})$ finitely generated by using the Schwarz-Milnor lemma
Recently, I have been studying the modular group $G=\mathrm{SL}(2,\mathbb{Z})$, and I am trying to prove $G$ is finitely generated by using the Schwarz-Milnor lemma in geometric group theory.I am ...
- 111
7
votes
0
answers
220
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Is there a Cayley graph with end space infinite and discrete?
A Cayley graph of a finitely generated group must be locally finite, and we know end spaces of locally finite graphs must be compact - so we can't have an infinite and discrete end space in this ...
- 407
12
votes
1
answer
323
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Does every mapping class group embed into some $\mathrm{Out}(F_n)$?
The title is pretty much the whole question. Let $S_g$ be a closed, oriented surface of genus $g$. Does there exist $n$ such that the mapping class group $\mathrm{Mod}(S_g)$ embeds as a subgroup of $\...
- 3,740
1
vote
0
answers
89
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The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results.
Let $A$ be a finitely generated abelian group,...
- 823
4
votes
1
answer
210
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Estimating an upper bound of hyperbolicity constants in Gromov-hyperbolic groups
Is there any good references with an explicit estimating hyperbolicity constants of hyperbolic groups? I was thinking about whether there is any relation between the relator of maximal length in the ...
- 81
8
votes
1
answer
349
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Finite two-relator groups and quotients of knot groups
Let $G$ be a one-relator group $\langle A \mid R = 1 \rangle$. Then clearly $G$ is finite if and only if it is cyclic of finite order, i.e. can be given by a presentation $\langle a \mid a^n = 1 \...
8
votes
1
answer
271
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$K$-theory and its dual
I am reading a paper which uses some $K$-homology which is the homology theory dual to $K$-theory can be defined using the homotopy theoretic formulation:
$$
K_\ast(X)\cong\pi_\ast(K\wedge X).
$$
...
- 81
5
votes
0
answers
87
views
Groups with a weaker (?) form of the Helly property
Let $G$ be a finitely generated group, with a fixed word metric $d$ coming from some finite generating set. Fix $n\in \mathbb{N}$, and suppose that for all finite $S\subseteq G$ with $\mathrm{diam}(S)=...
- 3,740
7
votes
1
answer
255
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Krasner–Kaloujnine universal embedding theorem for finitely generated groups?
The Krasner–Kaloujnine universal embedding theorem states that any group extension of a group $H$ by a group $A$ is isomorphic to a subgroup of the regular wreath product $A \operatorname{Wr} H$. When ...
- 775
3
votes
1
answer
100
views
Representation of Lie groups inducing a quasi-isometric embedding of their symmetric spaces
Let $G_{1}$ and $G_{2}$ be connected semisimple real Lie groups with no compact factors and finite center and let $K_{1}$ and $K_{2}$ denote some fixed choice of their maximal compact subgroups, ...
4
votes
1
answer
259
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Howson's property for amalgams of free groups
Recall that a group has the Howson property (also known as the finitely generated intersection property) if the intersection of any two finitely generated subgroups is again finitely generated.
I am ...
- 51
3
votes
1
answer
107
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Polynomial isoperimetric inequalities for finitely presented subdirect products of limit groups
Does every finitely presented subdirect product of limit groups admit a polynomial isoperimetric inequality? That is, does there exist a constant $C > 0$ and an integer $d \geq 1$ such that for ...
- 39
7
votes
2
answers
1k
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Groups killed by centralizing one element
What groups $G$ contains an element $g$ such that $G/(g\text{ is made central})=1$ (or equivalently $[g,G]=G$, where $[g,G]:=\langle[g,h]\colon h\in G\rangle$)?
A necessary condition is that $G$ is a ...
- 557
4
votes
1
answer
161
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Groups (not?) quasi-retracting onto $\mathbb{Z}$ via closest points projection
Inspired by this question we ask:
Suppose that $G$ is an infinite group. Suppose that $X$ is a finite generating set of $G$. Let $\Gamma = \Gamma(G, X)$ be the resulting Cayley graph. Does $\Gamma$ ...
- 28.2k
5
votes
1
answer
138
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Groups (not) quasi-retracting onto $\mathbb{Z}$
Let $X$ and $Y$ be metric spaces, with metrics $d_X$ and $d_Y$ respectively. A function $f\colon X\to Y$ is coarse Lipschitz if there exist constants $C,D>0$ such that $d_Y(f(x),f(x'))\le C d_X(x,x'...
- 3,740
8
votes
2
answers
352
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Embedding f.g. groups in 2-generated groups
Let $G$ be a finitely generated group. Can $G$ be embedded as a finite-index subgroup of a 2-generated group? 100-generated?
I strongly doubt it but I don't know a counterexample.
- 9,712
11
votes
1
answer
330
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What is the minimal genus of a surface acted on by the symmetric group $S_n$?
For $G$ a finite group, it is easy to construct a (connected, orientable) surface with a faithful action of $G$. E.g.: take a disjoint union of $G$ many spheres, and add a 1-handle for every edge in ...
- 43.2k
3
votes
1
answer
152
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$C'(\lambda)$ small cancellation for $\lambda < \frac{1}{6}$
I'm working with Gromov's density model of random groups, and a nice fact is that for a fixed density parameter $0 \leq d \leq 1$, a generic group in the density model satisfies the $C'(2d)$ small ...
- 495
2
votes
1
answer
202
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Strong Liouville property of virtually abelian groups
Let $G$ be a finitely generated group and let $\mu$ be a symmetric non-degenerate measure on $G$. By strong Liouville property for $(G, \mu)$, we mean that every positive $\mu$-harmonic function on $G$...
- 1,407
6
votes
1
answer
200
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Are finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ virtually special?
This might be a silly question--but are there any examples of finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ that are known to not be virtually special?
- 103
4
votes
0
answers
83
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Additive characters from coarse quotient maps
Let's consider a (finitely generated) group $\Gamma$ and a
coarse quotient map
$q\colon\Gamma\to\mathbb{R}$.
I'm interested in the 1-cocycle
$\sigma\colon\Gamma\to\ell_\infty\Gamma$,
defined by $\...
- 10.1k
3
votes
2
answers
468
views
How fast does the number of "fixed" points grow compared to the size of the ball in the following group?
I have copied this question from Math.StackExchange, in the hope that some experts here can provide some relevant insight.
Let $M = \oplus_{i\in \mathbb Z} V^{(i)}$ where each $ V^{(i)} \cong \mathbb ...
- 823
6
votes
1
answer
146
views
If $X$ is a hyperbolic, locally finite graph with $\partial X \cong S^1$, and $G$ acts cocompactly but not properly on $X$, what can we say?
It is an important and deep fact of geometric group theory that if the Gromov boundary of a hyperbolic group $G$ is a circle, then $G$ is virtually Fuchsian [Tukia, Gabai, Casson-Jungreis...]. I am ...
- 629
3
votes
0
answers
93
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References for variations of Seifert–van Kampen's theorem: HNN extensions and "sensible" intersections
A basic consequence of the Seifert–van Kampen theorem is the following.
Theorem: Consider a union of topological spaces $X$, $Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-...
- 1,033
11
votes
1
answer
250
views
Recognising the elements of the Grigorchuk group
The Grigorchuk group $\mathfrak{G}= \langle a,b,c,d \rangle$ is a group of automorphisms of the infinite rooted binary tree $\mathcal{T}_2$. Every element of $\mathfrak{G}$ can be represented by a ...
- 8,401
3
votes
1
answer
209
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A few points of clarification on the Martin boundary
Let $\Gamma$ be a finitely generated group, and let $M$ be the Martin boundary of $\Gamma$. I was reading the article on Martin boundary on Encyclopedia of Math, and I have a few questions about what ...
- 1,407
14
votes
2
answers
851
views
Examples of finitely presented subgroups of $\operatorname{GL}(n,\mathbb{Z})$ with unsolvable decision problems
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Does there exist a finitely presented subgroup of $\GL(n,\mathbb{Z})$ for which it is known that the conjugacy problem is unsolvable (if yes, ...
- 143
7
votes
2
answers
353
views
Finite normal subgroup of mapping class group
Let $\Sigma$ be a finite-type orientable surface with negative Euler characteristic, and $\mathrm{Mod}(\Sigma)$ denote the mapping class group. What are the finite normal subgroups in $\mathrm{Mod}(\...
- 605
0
votes
0
answers
138
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Model-theoretic construction of Gromov boundaries on groups
For context, I'm only a second year undergraduate mathematician, so I won't know much.
For third year, I'm hoping to do a research project. I met up with a professor who might be my supervisor today, ...
- 101
7
votes
1
answer
503
views
Are Artin-Tits groups ordered groups?
We consider Artin-Tits groups of two generators $(I_2(n))$. Are these groups ordered groups?
3
votes
2
answers
223
views
Measures with superexponential moments on finitely generated groups
Let $\Gamma$ be an infinite finitely generated group and let $\nu$ be a measure on $\Gamma$ which generates a transient random walk. I was reading this paper, and the authors prove many of their ...
3
votes
1
answer
123
views
Point stabilizers of the Floyd boundary of a group
Let $G$ be a finitely generated group. Consider the Floyd boundary as defined in https://www.unige.ch/math/folks/karlsson/free.pdf by A. Karlsson. For a Floyd function f, we denote the Floyd boundary ...
- 301
0
votes
0
answers
58
views
A stronger(?) notion than uniform contractibility
Let's call a metric space $ X $ strongly contractible if there exists a
function $ \rho : \mathbb{R}_+ \to \mathbb{R}_+ $ such that for every ball $
B(x;r) $ around a point $ x \in X $ we have:
$ B(x;...
- 169
0
votes
0
answers
125
views
When $G$ is amenable is true that for a set $A \in p$ which is left piecewise syndetic then $\{ x: Ax^{-1} \in p \}$ is both sided syndetic?
Let be $G$ be a discrete group. I recall the definition of syndetic sets, thick sets and piecewise syndetic sets.
Definitions:
A set $A$ is left syndetic if there exist a finite $H \subset G$ such ...
- 101
6
votes
1
answer
216
views
On intersection of finite index fully invariant subgroup
Let $G$ be a group. A subgroup $H$ of $G$ is said to be fully invariant if for every endomorphism $\phi $ of $G$, we have $\phi(H) \subseteq H$. For a finitely generated residually finite group $G$, ...
- 355
2
votes
0
answers
68
views
Amplification argument for hyperlinear groups
Let us define a group $G$ to be a hyperlinear group if it satisfies the conclusion of Theorem 3.6. in these notes by Vladimir Pestov. It is well-known that one can use the so-called amplification ...
- 6,214
2
votes
1
answer
97
views
Fundamental domain for the action on curve complex
Suppose $S$ is a surface and the Mapping Class Group, $Mod(S)$, of $S$, is the group of self-homeomorphisms of $S$, up to isotopy. This group acts on a graph called "curve graph", denote by $...
- 301
3
votes
1
answer
165
views
When the fundamental group of subgraph of groups embeds?
Given a connected graph of groups $\mathcal G$ (where edge maps are embeddings), by a subgraph we mean a graph of groups obtain by omitting some vertices, some edges, and replacing the remaining ...
- 1,338
4
votes
0
answers
79
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On the induction step in Theorem 2.6 of "Homological stability for linear groups" by Kallen
I am currently reading the proof of the connectivity theorem of Wilberd van der Kallen (Theorem 2.6 in https://link.springer.com/article/10.1007/BF01390018) for a seminar talk. I am a little stuck on ...
- 161
6
votes
1
answer
207
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Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces
I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
- 101
4
votes
1
answer
202
views
Green's kernel estimates on finitely generated groups
I was reading a paper by W. Hebisch and L. Saloff-Coste titled "Gaussian Estimates for Markov Chains and Random Walks on Groups" where I came to know about certain bounds on convolution ...
- 131
8
votes
4
answers
601
views
Residual finiteness of hyperbolic 3-manifold groups
So the consequence of the geometrization (according to 3-manifold group note) is that any finite-volumed hyperbolic 3-manifold is residually finite. So the question is:
Q1. If $M$ is an infinite-...
2
votes
0
answers
275
views
Interpretation of Kazhdan T property cohomologically
$\newcommand{\triv}{\mathit{triv}}$A group $G$ has property $T$ if $\triv$ is isolated in the space of unitary representations with the Fell topology.
In general, we heuristically have $H^1(G,Ad(V))$ (...
- 303
4
votes
0
answers
148
views
A question about Gromov's proof of a "more effective version of the main theorem"
In the paper "Groups of polynomial growth and expanding maps" Gromov proves the following "effective version of the main theorem"
For any positive integers $d$ and $k$, there ...
- 141
6
votes
1
answer
403
views
Do acyclic amenable groups exist?
Is there an example of a nontrivial discrete amenable group with vanishing integral homology?
To put the question in contrapositive. Given arbitrary acyclic group $Q$, is there some reason for the ...
- 4,600
0
votes
1
answer
198
views
Finding automorphism groups of regular graphs [closed]
Can some body help me with some source code for finding automorphism groups of regular maps?. For example: the type of graph is denoted as $\{p, q\}$, which means that they are tessellations of the ...
4
votes
0
answers
453
views
Problem 1.8 from Kirby's list
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 ...
