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
943
questions
12
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What is the least $n\ge1$ for which there is an $n$-dimensional closed flat manifold with perfect fundamental group?
In answer to the question "Is there a flat manifold with trivial first homology?" I proposed choosing a finite perfect group $P$ and a surjection $\phi:F\to P$ where $F$ is a free group of ...
7
votes
0
answers
232
views
In search of a quick proof that groups acting freely on $\mathbb R$-trees are linear
Many years ago I had the idea to use non-standard analysis to prove that a group acting freely on an $\mathbb R$-tree must be linear. The heuristic went like this:
A non-standard model $G^*$
of the ...
1
vote
1
answer
175
views
A question from the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel
In the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel, the author has made the following statement in the proof of Corollary D in page no. 18 ( https://arxiv.org/pdf/...
4
votes
0
answers
140
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A variation of Zuk's isoperimetric inequality for groups
$\DeclareMathOperator\diam{diam}\DeclareMathOperator\inrad{inrad}$There is a isoperimetric inequality (conjectured by Sikorav and proven by Żuk (Topology 39 (2000) 947–956) which holds in every Cayley ...
7
votes
1
answer
216
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If the number of ends of Freudenthal space is infinite, then its space of ends is homeomorphic to the Cantor set?
I don't know whether this is the right place to discuss a part of someone's thesis or not. If it is wrong, let me know; I will delete my post.
I am reading this thesis.
Corollary 4.1.15. on page 63 ...
1
vote
1
answer
160
views
What properties are preserved by quasi-isometries
Recently, I came across the notion of quasi-isometries, while thinking of "discrete spaces which are surrogates for approximate continuous ones".
What (metric)/geometric properties are ...
6
votes
0
answers
338
views
Arithmetic Teichmüller curves, first eigenvalue of the Laplacian, McMullen's expander conjecture
$\DeclareMathOperator\SL{SL}$By a result due to Ellenberg and McReynolds, any finite index subgroup $\Gamma$ of $\Gamma(2) \subset \SL\left(2,\mathbb{Z}\right)$ is the Veech group of an arithmetic ...
1
vote
0
answers
69
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Another matrices for a semigroup with intermediate growth
Nathanson showed that the Okninski's semigroup $S$ of $2×2$ matrices which is generated by the set $H=\{A,B\}$, where
$
A=\begin{bmatrix}
1&1\\
0&1\\
\end{bmatrix}
,
B=\begin{bmatrix}
1&0\\...
2
votes
0
answers
141
views
"Hyperbolic rigidity of higher rank lattices" by Thomas Haettel
$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\CC{C}$In the article "Hyperbolic rigidity of higher rank lattices", Thomas Haettel has proved the following theorem: Let $\Gamma$ be a ...
4
votes
1
answer
497
views
Amenable subsets of groups
Let $X$ be a subset of a group $G$. We say that $X$ is left amenable with respect to $G$ if there is a function $\mu:\mathcal P(G)\to [0,\infty]$ with the following three properties.
$\mu(A\cup B)=\...
10
votes
1
answer
138
views
Iterated algebraic fibering
A finitely generated group $G$ algebraically fibers if there is an epimorphism $G\to\mathbb{Z}$ with finitely generated kernel. Since this kernel is finitely generated, we can ask whether *it* ...
4
votes
0
answers
74
views
Counting the number of free bases of $F_n$ with elements of bounded length
Let $F$ be a free group of finite rank and fix a free generating set $X$ of $F$. Let $P_r$ denote the set of all free generating sets of $F$ whose elements have length bounded by $r$ (when considered ...
3
votes
1
answer
285
views
Can graphs of groups be thought of as "graph objects" in the category of groupoids?
An undirected graph is sometimes defined as a pair of sets $V$ and $E$ (vertices and oriented edges), together with two maps $i,f: E\to V$ (sending a directed edge its initial/final vertex) and a map $...
5
votes
1
answer
303
views
In what sense is Bass-Serre theory the one-dimensional version of orbifold theory
The Wikipedia article on Bass-Serre theory claims that graphs of groups (in the context of Bass-Serre theory) "can be viewed as one dimensional versions of orbifolds." I hazily see a ...
4
votes
1
answer
182
views
Infinitely divisible elements in Gromov hyperbolic groups
An element $g\in G$ in a group $G$ is called infinitely divisible if $b=y^n$ for infinitely many different $n\in {\Bbb Z}$. It is not hard to find a finite CW-complex (or even a compact manifold) ...
2
votes
0
answers
100
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Definition of the category QMet of metric spaces and quasi-isometries
I am following Clara Löh's Geometric Group Theory. An Introduction, and in remark 5.1.12, she defines the category QMet whose objects are metric spaces and whose morphisms are quasi-isometric ...
2
votes
1
answer
461
views
Growth rate of an outer automorphism of a free product
$\DeclareMathOperator\Out{Out}$Let $G=G_1\ast\cdots\ast G_k\ast F_p$ be a Grushko decomposition of a finitely generated group $G$, $\mathcal{O}$ the outer space relative to this decomposition, $[\phi]\...
2
votes
1
answer
133
views
Subgroup growth of direct product
I have started reading about subgroup growth and, to my surprise, I haven't found a reference to whether direct products preserve subgroup growth.
Recall that, given a finitely generated group $G$, ...
0
votes
0
answers
58
views
How large must "weak Besicovitch" subsets of groups be?
Consider a group $G$; let call $A\subset G$ a weak Besicovitch subset whenever every element of $G$ can be written under the form $gh^{-1}$, where $g,h\in A$.
General question: how large must a weak ...
3
votes
0
answers
95
views
Order type of monotone functions on $\Bbb N$ up to affine conjugation
Let's introduce order on non-strictly monotone functions $\Bbb N \to \Bbb N$ such that $f \leq g$ if $f(n) \leq Cg(Cn + C) + C$ and, of course, identify such $f, g$ if $f \leq g \leq f$. (Note absence ...
3
votes
0
answers
126
views
the growth rate of poly-$\mathbb{Z}$ group
I am interested in the growth rate of the poly-$\mathbb{Z}$ group. Let $G$ be a poly-$\mathbb{Z}$ group, i.e $$G =(\dots((\mathbb{Z} \rtimes_{\phi_1} \mathbb{Z})\rtimes_{\phi_2} \mathbb{Z}) \rtimes_{\...
11
votes
1
answer
549
views
If $(F_n)_n$ is a Følner sequence satisfying Tempelman's condition, is $(F_n^{-1}F_n)_n$ also Følner?
Let $G$ be a countable group. A Følner sequence is a sequence of finite subsets $(F_n)_n$ such that
$$\lim_{n\to\infty} \frac{|KF_n \mathbin\triangle F_n|}{|F_n|} = 0$$
for each fixed finite subset $K ...
7
votes
1
answer
153
views
Density of “diagonal sets” in amenable groups
Let $G$ be a countable amenable group with a (left) Følner sequence $(F_n)$. Let $\Gamma$ be a subset of $G$ that has density $1$ with respect to $(F_n)$, in the sense that
$$
\lim_{n \to \infty} \...
4
votes
0
answers
193
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 $\phi$...
8
votes
1
answer
395
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 ...
4
votes
1
answer
371
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 ...
1
vote
1
answer
73
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 ...
4
votes
0
answers
107
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 ...
4
votes
1
answer
254
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 ...
8
votes
1
answer
201
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 ...
9
votes
4
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, ...
3
votes
2
answers
182
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 ...
10
votes
2
answers
738
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 ...
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 (...
2
votes
1
answer
197
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 ...
4
votes
0
answers
119
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 ...
4
votes
1
answer
113
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
213
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:...
1
vote
0
answers
91
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 ...
13
votes
2
answers
606
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 ...
7
votes
0
answers
165
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{...
3
votes
1
answer
165
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 &...
8
votes
0
answers
374
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-...
1
vote
1
answer
177
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\...
4
votes
0
answers
196
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 ...
8
votes
1
answer
153
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?
...
3
votes
0
answers
368
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 (...
0
votes
0
answers
102
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)....
4
votes
0
answers
230
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 ...
8
votes
2
answers
763
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 ...