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
822
questions
4
votes
0
answers
53
views
Density of ``diagonal sets'' in amenable groups
Let $G$ be a countable amenable group with a (left) Følner sequence $(F_n)$. Let $\Gamma$ be a subset of $G$ that has density $1$ with respect to $(F_n)$, in the sense that
$$
\lim_{n \to \infty} \...
- 291
3
votes
0
answers
102
views
Does there exist a finitely generated, torsion group $G$ with a residually finite ascending HNN extension?
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can provide me some insight.
Let $G$ be a group with an injective endomorphism $\phi$...
- 293
5
votes
1
answer
179
views
When are groups generated by reflections in a triangle discrete?
Take a triangle in the (Euclidean or hyperbolic) plane, and consider the group of isometries generated by the reflections in the three sides of the triangle. If the angles between adjacent sides are ...
- 731
3
votes
0
answers
139
views
When is a generalised Baumslag-Solitar group linear?
$\DeclareMathOperator\BS{BS}$The linearity of the Baumslag-Solitar groups $\BS(m, n)=\langle a, t\mid t^{-1}a^mt=a^n\rangle$ is completely understood, and it may be phrased as: $\BS(m, n)$ is linear ...
- 2,124
1
vote
1
answer
57
views
Let $H$ be a co-dimension 1 quasiconvex subgroup of a one-ended hyperbolic group $H$. Does $H$ permute the components of $\partial G - \Lambda H?$
Let $H$ be a co-dimension 1 quasiconvex subgroup of a one-ended hyperbolic group $G$. In particular, $\partial G$ is connected but $\partial G - \Lambda H$ is disconnected. The number of components of ...
- 435
4
votes
0
answers
86
views
Hilbert space compression of CAT(0) groups
Does there exist a CAT(0) group with Hilbert space compression $<1$?
The Hilbert space compression of a metric space $(X,d)$, e.g. a group endowed with the word metric given by a finite generating ...
- 6,393
3
votes
1
answer
148
views
Residual finiteness of random groups with property (T)
A well known result of A. Zuk states that for $\frac{1}{3} < d < \frac{1}{2}$, a random group $\Gamma$ with respect to Gromov's density model with density $d$ has Kazhdan's property (T) with ...
- 237
8
votes
1
answer
160
views
Existence of properly discontinuous and cocompact action
Let $M$ be a complete Riemannian manifold. My question is, under what conditions does $M$ admit a discrete group of isometries $\Gamma$ which acts properly discontinuously and cocompactly on $M$, that ...
- 81
8
votes
3
answers
1k
views
Proving that a countable group is not finitely generated
I would like to learn about techniques for proving that a countable group is not finitely generated. I am also interested in learning about examples. Finally, I am particularly, but not exclusively, ...
- 101
2
votes
2
answers
129
views
HNN decomposition of finite rank free group over infinite rank subgroups
It's a nice result of Swarup that whenever a free group $G$ splits as an HNN extension $G = J \ast_{H,t}$ with $H$ a finitely generated subgroup, there exist splittings $J = J_1 \ast J_2$ and $H = H_1 ...
- 200
10
votes
2
answers
496
views
Examples of hyperbolic groups with non-hyperbolic subgroups
In a previous question, I asked about hyperbolic groups in which every finitely generated subgroup is hyperbolic. I am now curious about the reverse question: what are some examples of hyperbolic ...
- 373
17
votes
3
answers
1k
views
Examples of locally hyperbolic groups
It is well-known that a subgroup of a hyperbolic group need not be hyperbolic. Let us say that a (finitely generated) group $G$ is locally hyperbolic if all its finitely generated subgroups are (...
- 373
2
votes
1
answer
176
views
Examples of group families with solvable uniform word problem
I would like to know of any examples of families of groups that are known (or conjectured) to have a solvable uniform word problem, i.e. an algorithm that given a presentation $P$ of a group in the ...
- 1,294
4
votes
0
answers
90
views
Type $F_n$ and quasi-isometry with highly connected space
Recall that a group is of type $F_n$ if it has a classifying space with finite $n$-skeleton. For example type $F_1$ means finitely generated, and type $F_2$ means finitely presented.
Question: For a ...
- 2,537
4
votes
1
answer
78
views
Looking for a citation: the Rips $n$-complex of a $\delta$-hyperbolic group is contractible for high enough $n$
Given a $\delta$-hyperbolic group $G$, I have been told that the Rips $n$-complex of $G$ is contractible for high enough $n$. The only proof I have found for this statement is in an expository essay ...
1
vote
0
answers
107
views
Example of CAT($k$) space [closed]
Good time of day. I repeat the question from MSE (https://math.stackexchange.com/questions/4464888/question-about-example-of-catk-space) because no response has been received.Question is the following:...
- 77
1
vote
0
answers
68
views
Question about coarse fixed point property in large-scale geometry
I read the article of Steven Hair "A degree-theoretic proof of a coarse fixed point principle". I have the following question.
I start with some main definitions from this article. A coarse ...
- 77
13
votes
2
answers
490
views
Mapping Out(F_n) to the mapping class group
Let $\mathrm{Out}(F_g)$ denote the automorphism group of a free group, and $\mathrm{Mod}_g$ the mapping class group of a closed oriented genus $g$ surface. Is there a map, as indicated with the dashed ...
- 37.5k
7
votes
0
answers
124
views
Reduced group C*-algebra $C^*_r(\mathbb{Z}/2*\mathbb{Z}/2)$: norm of specific elements
Consider the free product of $\mathbb{Z}/2$ with itself with generators
$$
\mathbb{Z}/2*\mathbb{Z}/2=\langle u,v\mid u^2=1=v^2\rangle
$$
and regard its group $C^*$-algebra
$$
C^*(\mathbb{Z}/2*\mathbb{...
- 598
3
votes
1
answer
89
views
Carne-Varopoulos bound and stationary measure
Let $\Gamma$ denote the Cayley graph for a finitely generated group $G$, and let $p_n(x, y)$ denote the transition probability that a random walk starting at $x$ reaches $y$ at time $n$. A famous &...
- 31
7
votes
0
answers
279
views
The figure eight knot complement in $S^3$
Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. Exercise 5.4 (on page 101) gives us a presentation of the fundamental group of $S^3 - K$ where $K$ is the figure-...
- 71
1
vote
1
answer
143
views
Finiteness of $\ell^2$-Betti numbers
I'm reading the paper "Improved algebraic fibering" by Sam Fisher (https://arxiv.org/pdf/2112.00397.pdf) and in the proof of lemma 6.4 it claims the followng:
$(\mathcal{D}_{\mathbb{F}K}\ast\...
- 111
4
votes
0
answers
149
views
Distortion in the Brin-Thompson 2V
Is it known whether the Brin-Thompson 2V contains a distortion element? By this I mean an element $f$ such that the word norm $|f^n|$
grows sublinearly, and $f$ is of infinite order. If such an ...
- 4,631
8
votes
1
answer
112
views
Are the non-free factors of Grushko decomposition of a finitely generated convex-cocompact (but not cocompact) subgroup of PSL$(2,\mathbb{R})$ finite?
See Grushko decomposition theorem.
Are the non-free factors of Grushko decomposition of a finitely generated convex–cocompact (but not cocompact) subgroup of $\operatorname{PSL}(2,\mathbb{R})$ finite?
...
- 135
3
votes
0
answers
305
views
What about a Cayley n-complex for n>2?
Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...
- 23.3k
0
votes
0
answers
80
views
Inverse limit in category of graphs
Let $... \leftarrow G_i \leftarrow G_{i+1} \leftarrow ...$ be an inverse system of graphs (which might not be locally finite), morphisms are symplicial maps (we allow collapsing of edges to vertices)....
- 1
4
votes
0
answers
111
views
Convex core and geometric finiteness of negatively curved manifolds
I am reading a paper on hyperbolic geometry where they are using the concept of "convex core" in the context of "geometric finiteness". Roughly, this means (from Definition F4 of a ...
- 41
7
votes
2
answers
549
views
Group of exponential growth always contains a free sub-group?
I am not very conversant with the growth of a group, so this may be a very silly question.
It is known that $F_2$, the free group of rank $2$, has exponential growth. I was wondering whether the ...
- 931
4
votes
0
answers
98
views
Ends of a negatively curved Riemannian manifold
Let $M$ be a complete Riemannian manifold. Let us use the standard definition of "end", for example, as in this article. If $M$ has non-negative Ricci curvature, it is well-known that it has ...
- 41
7
votes
4
answers
391
views
Is every virtually free group residually finite?
Question: Is every (finitely generated) virtually free group residually finite?
A well-known question asks whether every hyperbolic group is residually finite (Mladen Bestvina. Questions in geometric ...
3
votes
0
answers
45
views
Decomposition about splitting of symmetric spaces of compact type
I get stuck in the following question:
Why does a locally symmetric space of compact type $M$ split locally irreducible components of dimension $\geq 2$ which are Einstein? In particular, why are all ...
- 445
5
votes
1
answer
294
views
Is this semi-direct product residually finite?
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can help me find a way to check the residual finiteness of this group.
Consider the ...
- 293
29
votes
1
answer
1k
views
Group theory with grep?
While reading Bill Thurston's obituary in the Notices of the AMS I came across the following fascinating anecdote (pg. 32):
Bill’s enthusiasm during the early stages of mathematical discovery was ...
- 291
1
vote
1
answer
89
views
Number of orbits for abelian group actions
Suppose $G$ is an abelian group acting faithfully on two sets, $X$ and $Y$, of the same size. None of $G$, $X$ and $Y$ is finite.
Now suppose $G$ is the union of abelian groups $G_i$, where $i$ varies ...
- 3,371
2
votes
1
answer
157
views
Parahoric subgroup over a local field
$\DeclareMathOperator\SL{SL}$Let $F$ be a local field and $\mathcal{O}_{F}$ its valuation ring. Let $\pi\in \mathcal{O}_{F}$ be a uniformizer and $\mathfrak{p}=\pi\mathcal{O}_{F}$. Let $G$ be a split ...
- 419
11
votes
1
answer
426
views
Counterexamples to an analog of Cannon's Conjecture which do not arise from manifolds?
Let $\Gamma$ be a finitely presented hyperbolic group with boundary homeomorphic to $S^{n-1}$. Are there any examples of such $\Gamma$ which are known to not be the fundamental group of any $n$-...
- 317
4
votes
1
answer
234
views
Permuting subgroups with the same finite index
Suppose that we have a finitely generated residually finite group $G = \langle g_1,\ldots,g_r \rangle$ with polynomial growth. Let $H$ be a subgroup of $G$ with finite index $m$. Let $\phi$ be an ...
- 293
6
votes
1
answer
132
views
Positive harmonic functions on nilpotent groups & Random walk on groups with a finite number of generators
I want to read the following papers in the English version which I could not find anywhere (the only papers I can get are the Russian versions). Kindly help me out.
Gregory A. Margulis, Positive ...
- 931
8
votes
3
answers
387
views
Spectral radius of a finitely generated group
Let $G$ be a finitely generated group and $\Gamma$ be its Cayley graph with the usual word metric. Let $\mu$ be a symmetric non-degenerate measure on $G$ (maybe with finite support or smooth), and ...
- 1,081
5
votes
2
answers
224
views
If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, must the fixed points of $1_\mathbb{Z}$ be a finitely generated group?
I've copied over this question from what I asked on StackExchange, in the hope that an expert here can readily answer the question.
Is there an example of a group $G=K\rtimes \mathbb{Z}$ satisfying ...
- 171
7
votes
1
answer
111
views
Discrete cocompact group of isometries of Nil
Is it true that every group quasi-isometric to the Heisenberg group admits a proper cocompact action by isometries on Nil?
- 412
5
votes
1
answer
109
views
Given a quasi-convex subgroup $H$ of hyperbolic $G$, can we decide if two elements $x,y \in G$ lie in the same double coset of $H$?
I've come across the following question in my research, which seems elusive but is almost surely decidable.
Let $H$ be a quasi-convex subgroup of the hyperbolic group $G$. Given $x, y \in G$, we wish ...
- 435
5
votes
1
answer
85
views
Infinite groups that admit a discrete, co-compact, bilipschitz action on $\mathbb{R}^3$
Apart from the abstract types of the crystallographic groups, are there any other abstract groups that admit a proper, co-compact, uniformly bilipschitz action on $\mathbb{R}^3$?
- 412
3
votes
0
answers
63
views
First Betti number of finitely non-co-hopfian groups
Let $G$ be a finitely generated group. Assume that $G$ is a finitely non-co-hopfian group in the sense that there is a group embedding $i: G\hookrightarrow G$ such that $1<[G\colon i(G)]<\infty$...
- 497
5
votes
2
answers
739
views
Potential counterexamples to Bass' trace conjecture
Motivation: The following is a theorem of Berrick-Hesselholt (essentially also due to Linnell, though not in this form):
Let $G$ be a group. Suppose that for every subgroup of $G$ isomorphic to $\...
- 8,945
3
votes
1
answer
90
views
Are geometric actions on CAT(0) spaces with isolated flats minimal on the boundary?
Suppose $X$ is a $CAT(0)$ space with isolated flats, $\partial X$ its visual boundary and $G$ acts properly discontinuously amd cocompactly on $X$. Must the $G$ action on $\partial X$ have a dense ...
- 211
0
votes
1
answer
170
views
Torsion-free subgroup of affine group
Let $G$ be a finitely generated group and $\varphi:G\to \operatorname{Aut}(\mathbb C)$ a homomorphism, where $\operatorname{Aut}(\mathbb C)$ is the group of complex affine transfromations $a z+b$.
Can ...
- 185
5
votes
0
answers
103
views
Finitely presentable group with purely infinite full group $C^*$-algebra?
Does there exist an example of a finitely presentable group whose full group $C^*$-algebra is purely infinite,
resp. is it known to be impossible?
- 598
8
votes
1
answer
141
views
Non-finitely presented FP groups with cohomological dimension $2$
In this recent preprint, the authors construct a certain uncountable family of non-finitely presented FP groups. Recall that group is an FP group if the trivial $\mathbb Z[G]$-module $\mathbb Z$ has a ...
- 8,945
1
vote
1
answer
185
views
Which properties can be read off the balls of a Cayley graph?
For which properties (P) [of groups] does the following hold:
given a group $G$ which has a finite presentation with at most $n$ relations of length at most $\ell$, there is a $R(n,\ell)$ so that, if ...
- 4,244