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
948
questions
12
votes
1
answer
489
views
Is the Thompson group F locally indicable?
A group $G$ is called locally indicable if for any finitely generated subgroup $H \subset G$, there is a non-trivial homomorphism from $H$ to the real additive group $(\mathbb{R},+)$.
Is the Thompson ...
12
votes
1
answer
576
views
Is residual finiteness a quasi isometry invariant for f.g. groups?
A "residually finite group" is group for which the intersection of all finite index subgroups is trivial. Suppose $G$ and $G'$ are two quasi-isometric finitely generated groups. Does the residual ...
12
votes
2
answers
357
views
Are finitely generated amenable groups positively finitely generated?
Let $G$ be a finitely generated amenable group.
Is there a positive integer $n$ such that $n$ random elements of $G$ generate it with positive probability?
Being more formal, note that $G^n$ is ...
12
votes
1
answer
849
views
Isometries of some simple Cayley graphs
Consider a Cayley graph of a group $G$ with respect to a symmetric finite generating set $S$.
There are some obvious candidates to isometries of this graph - for example, translation by elements of $G$...
12
votes
1
answer
515
views
What are smallest finite images of triangle groups?
Given integers $a, b, c$ all at least 2, I would like to identify the smallest group with an element $x$ of order $a$, element $y$ of order $b$ such that the product $yx$ has order $c$. For example, ...
12
votes
1
answer
377
views
Do finitely generated groups of polynomial growth satisfy a "uniform covering property?"
Let $G$ be a finitely generated discrete group with a finite symmetric generating set $S=S^{-1}\subset G$. For every group element $g$, define $\|g\|_S$ to be the length with respect to $S$, i.e. the ...
12
votes
1
answer
379
views
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 ...
12
votes
1
answer
305
views
Dualizing module for $\operatorname{Aut}(F_n)$
In The complex of free factors of a free group (pdf at Hatcher's page), Hatcher and Vogtmann defined a simplicial complex $FC_n$ called the ``complex of free factors'' of the free group $F_n$. They ...
12
votes
1
answer
432
views
Quasimorphisms and Bounded Cohomology: Quantitative Version?
Consider maps from a discrete group $\Gamma$ to the additive group $\mathbb{R}$. A function $f:\Gamma \to \mathbb{R}$ is called a quasimorphism if it is locally close to being a group homomorphism. ...
12
votes
2
answers
3k
views
Examples of "Monster" groups
I am planning a talk for a general graduate student audience. The topic is exotic examples of countable discrete groups ("monsters"). Some examples of properties that I'm interested in are:
1.) Non-...
12
votes
1
answer
1k
views
What is the Status of Borel conjecture today?
Let me recall the conjecture: $M$ and $N$ two aspherical closed $n$-manifolds with isomorphic fundamental groups, then $M$ and $N$ are homeomorphic.
12
votes
1
answer
266
views
Artin groups of type $D_n$ as mapping class groups?
According to Allcock (Braid Pictures for Artin groups, https://arxiv.org/abs/math/9907194), the Artin group $A(D_n$) of type $D_n$ may be realized as an index 2 subgroup of the orbifold fundamental ...
12
votes
1
answer
293
views
Hyperbolic 3-manifold groups acting on the plane
Can the fundamental group of a closed hyperbolic 3-manifold act freely on the plane by homeomorphisms? Freely and cocompactly? Freely, cocompactly, and preserving orientation?
12
votes
0
answers
457
views
Writing an element of a free product of $C_2$'s as a product of order-$2$ elements
My question is simple: Suppose that $G$ is isomorphic to the free product of finitely many copies of $C_2$. Is it true that any element $g \in G$ can be written as a product $g = s_1 \dotsm s_m$ such ...
12
votes
0
answers
336
views
Does Thompson's group $V$ have property AP?
Property AP: A discrete group $\Gamma$ has property AP (Approximation Property) if there exists a net $(\phi_i)_{i \in I}$ of finitely supported functions on $\Gamma$ such that $\phi_i \to 1 $ weak$^*$...
12
votes
0
answers
388
views
What is an example of a word hyperbolic group without a finite complete rewriting system?
I believe that it was an open question back when I was a graduate student whether every word hyperbolic group admits a finite complete (=Church-Rosser=Noetherian+confluent) rewriting system for some ...
11
votes
4
answers
1k
views
What are the most general classes of simplicial complexes or posets for which the Charney-Davis conjecture is known, and what is the most general setting for which it might expected to be true?
What I would like to know is exactly what the title asks:
What are the most general classes of
simplicial complexes or posets for
which the Charney-Davis conjecture is
known, and what is the ...
11
votes
4
answers
926
views
Finite subgroups of relatively hyperbolic groups
It is well known that in a given $\delta$-hyperbolic group there are only finitely many conjugacy classes of finite subgroups. This is clearly false for relatively hyperbolic groups since we have no ...
11
votes
2
answers
608
views
Constraints on the homology of amenable groups
Edit (March 24): My first question has been answered nicely, but I am still looking for an answer to the second one.
Due to the Kan–Thurston theorem, the homology of an arbitrary group can be anything ...
11
votes
3
answers
682
views
Embedding groups into groups with some vanishing homology groups
Which finite subsets $S \subset \mathbb{N}$ have the following property : every countable group $G$ embeds into a finitely generated group $\Gamma$ such that $H_i(\Gamma;\mathbb{Z})=0$ for all $i \in ...
11
votes
2
answers
721
views
Quasi-isometric rigidity of Nil
Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as ...
11
votes
2
answers
930
views
Subtle point in definition of BNS invariant
Let $G$ be a finitely generated group. Let $S(G)$ be the quotient of $\text{Hom}(G,\mathbb{R}) \setminus \{0\}$ by the equivalence relation that identifies two homomorphisms if they differ by scaling ...
11
votes
1
answer
604
views
Conjugating a subgroup of a group into a proper subgroup of itself
The following question came up in the class I'm teaching right now. There definitely exist groups $G$ with subgroups $H$ such that there exists some $g \in G$ such that $g H g^{-1}$ is a proper ...
11
votes
3
answers
344
views
Right-angled Artin groups that split as direct products
For a finite graph $X$, let $A_X$ denote the associated right-angled Artin group. Thus $A_X$ is generated by the vertices of $X$ subject to the relations $[v,w]=1$ whenever vertices $v$ and $w$ are ...
11
votes
1
answer
472
views
How to construct a group with specified growth function
Are there any procedures which given a nonnegative nondecreasing function on the integers will construct a finitely generated group with the same growth up to the usual equivalence of growth functions?...
11
votes
1
answer
329
views
Growth of Poincaré duality groups
Can one prove that Poincaré duality groups cannot have intermediate growth?
11
votes
1
answer
553
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 ...
11
votes
1
answer
888
views
Normal closures of finitely generated subgroups of a free group
Is it true that for every finitelty generated subgroup $H$ of infinite index in a free
group $F$ on the two letters $\{x,y\}$, there exists a finite index
subgroup $K$ of $H$, such that the normal ...
11
votes
1
answer
495
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 $n$-...
11
votes
1
answer
2k
views
Candidates for non-sofic groups
What are the "simplest" examples of countable groups that are not known to be sofic?
11
votes
1
answer
603
views
Analogues of the curve complex for Out(F)
Let $F$ be a finitely generated free group.
Question: Is there an authoritative survey of analogues of the curve complex for $\mathrm{Out}(F)$? If not, as seems likely, would a passing expert be ...
11
votes
1
answer
380
views
Embeddings of finitely generated groups into uniformly convex Banach spaces
de Cornulier, Tessera, and Valette (Geom. Funct. Anal. 17 (2007), 770-792) conjectured that a finitely generated group $G$ with its word metric admits a bilipschitz embedding into a Hilbert space if ...
11
votes
1
answer
358
views
Are groups with the Haagerup property hyperlinear?
In his 2008 paper Hyperlinear and Sofic Groups: A Brief Guide, Pestov asked (Open Question 9.5) whether every group with the Haagerup property is hyperlinear (or sofic). Has this question been ...
11
votes
1
answer
1k
views
Topological HNN extensions
First, let me recall what an abstract HNN extension is. Let $G$ be an abstract group, $A, B < G$ be subgroups of $G$ and $\phi : A \to B$ be an isomorphisms. Then there is a group $H$ and an ...
11
votes
2
answers
650
views
Quantum Cellular Automata on Riemannian manifolds and geometric group theory
We try to motivate our question. We have a certain logical/operational structure that has an
emergent physical interpretation. We are giving this structure a geometric setting via
quasi-isometries. ...
11
votes
1
answer
526
views
Is there a faithful transitive locally finite action of the modular group?
Is there a faithful transitive action of $G = \mathrm{PSL}_2(\mathbb{Z})$ on $\mathbb{Z}$ such that orbits under each $g \in G$ are finite?
11
votes
1
answer
526
views
Partial word orders on groups
This is a followup question related to this question. Recall that a left-invariant partial order on a finitely generated group $G$ is called a partial word order if for every $a\le b\le c$ we have $|...
11
votes
0
answers
313
views
Status of questions in "Group Actions on $\mathbb{R}$-trees"?
Culler and Morgan's "Group Actions on $\mathbb{R}$-trees" lists four questions at the end of the introduction. A few have been famously resolved by work of Rips, Bestvina–Feighn and others.
I'm ...
11
votes
0
answers
363
views
Amalgamated product of automatic groups
In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic?
Is this ...
11
votes
0
answers
415
views
Other than the Higman group, what other candidates do we have for non-sofic groups?
I know that the Higman group is the most widely studied candidate right now, but what are the others? For example, is (are) Thompson's group(s) sofic? And what about the Burger-Mozes groups? I haven't ...
11
votes
0
answers
668
views
Uniquely geodesic groups
Definition : A group is CAT(0) if it acts properly, cocompactly and isometrically on a CAT(0) space.
Examples : see this blog.
Remark : A CAT(0) space is uniquely geodesic, but the converse is ...
11
votes
0
answers
677
views
Definition of a uniformly bounded dual of a group
The unitary dual of a group $G$ is the set of equivalence classes of irreducible unitary representations of $G$ with the Fell topology. (This topology is defined using convergence of positive definite ...
10
votes
2
answers
2k
views
When is a Baumslag-Solitar group linear?
The Baumslag-Solitar group $BS(m,n)$ is given by the group presentation
$BS(m,n)=(a,b|ba^{m}b^{-1}=a^{n})$. When does it embed into a linear group? Thanks!
10
votes
2
answers
1k
views
Kazhdan's property (T) vs. residual finiteness
I have asked this question already on mathstackexchange but got no answer (see https://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...
10
votes
3
answers
1k
views
A finite index subgroup of the Mapping Class Group
Let $G$ be the mapping class group of a closed surface $S_{g}$. Bestvina-Bromberg-Fujuwara https://arxiv.org/abs/1006.1939 recently constructed a finite index subgroup $B$ of $G$ such that for every ...
10
votes
2
answers
437
views
Sequence of epimorphisms of residually finite groups stabilizes
Let $G_1 \to G_2 \to \cdots$ be a sequence of epimorphisms of finitely generated residually finite groups. Does it eventually stabilize? That is, are all but finitely many epimorphisms actually ...
10
votes
2
answers
748
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 ...
10
votes
4
answers
576
views
Is the action of $G$ on $H_1(T^n, \mathbb{Z}) = \mathbb{Z}^n$ faithful?
Let $G$ be a finite group of diffeomorphisms of the torus $T^n$ fixing some point $p$, i.e. $p$ is fixed by every element of $G$. I have two questions.
Is the action of $G$ on $H_1(T^n, \mathbb{Z}) = ...
10
votes
1
answer
360
views
Translation lengths in CAT(0) spaces
Let $a,b$ be two loxodromic isometries of a CAT(0) space. Assume that, for every $n \geq 1$, $a^nb$ is also loxodromic. Is it possible for the translation length of $a^nb$ to be bounded independently ...
10
votes
4
answers
881
views
Stallings' Theorem for free products of groups
There is a well known theorem which states that:
Theorem(Stallings):
For any immersion $f$ from a finite graph $D$ to $G$ there is a finite-sheeted covering space $D '$ of $G$ that extends $f$. ...