22
votes
Recognizing free groups
As indicated in the comments, it's undecidable in general to take as input a finite presentation of a group and try to output whether or not this group is free or not. This is a direct consequence of ...
22
votes
Accepted
Group generated by two irrational plane rotations
The commutator of any two elements of your group is a translation, so they all commute. So for example if $a$ and $b$ are two elements of your group then $[a,b]$ commutes with $[a^b,b]$. This is a ...
- 11.8k
18
votes
Accepted
Universal graph
I think that the answer is negative.
Assume that such graph $G$ on the vertex set $\{v_1,v_2,\ldots\}$ exists. We construct our not-embeddable graph $H$ on $\{1,2,\ldots\}$ by steps. On the $i$-th ...
- 103k
16
votes
dichotomy in hyperbolic groups
No, there is no such dichotomy. If $G$ is an infinite group with Property (T) and $H$ is any non-trivial group, then $G*H$ has neither of the two properties. This is because groups with Property (T) ...
- 2,941
15
votes
Classes of groups with polynomial time isomorphism problem
A two-generator, one-relator group with torsion is a group with presentation of the form $\langle a, b\mid R^n\rangle$, $R\in F(a, b)$ and $n>1$. Their isomorphism problem is decidable in quadratic ...
- 2,762
13
votes
Accepted
Morse theory on outer space via the lengths of finitely many conjugacy classes
You don't misunderstand, it's a subtle point that I'm sure I'll get wrong here too. You might find the proof of a slightly more general statement in Krstić and Vogtmann's "Equivariant Outer Space ...
- 1,986
13
votes
Accepted
Dehn functions of finitely presented simple groups
To answer the vaguer question: I think there is no known bound on the Dehn functions of finitely presented simple groups. Recall:
Boone–Higman Embedding Theorem.
A finitely presented group has ...
- 2,941
12
votes
Accepted
Problem 3.14 from Kirby's list
This problem is answered in the literature, with a caveat.
As indicated in the comments, it follows from the orbifold theorem + geometrization conjecture (to handle the case of orbifolds without fixed ...
- 66.8k
12
votes
Groups acting on infinite dimensional CAT(0) cube complex
Natural examples of finitely generated groups acting properly on median graphs of infinite cubical dimension (or, if you prefer, on infinite-dimensional CAT(0) cube complexes) include:
Thompson's ...
- 7,511
11
votes
Accepted
Examples of groups that are unknown to be acylindrically hyperbolic
Here is a list of groups for which acylindrical hyperbolicity is known or at least understood in some cases:
The iconic examples are mapping class groups of non-sporadic surfaces of finite type and ...
- 7,511
11
votes
Accepted
Analogous results in geometric group theory and Riemannian geometry?
I think Cheeger's inequality is a good example.
Riemannian geometry version
Let $M$ be a closed Riemannian $n$-manifold. Say that a $n-1$ dimensional submanifold $N$ separates $M$ if the complement of ...
- 28.8k
11
votes
Accepted
Does every sequence of group epimorphisms (between finitely generated groups) contain a stable subsequence?
Yes. This is just (metrizable) compactness in the space of normal subgroups of $G$. It is enough to assume that $G$ is countable (finitely generated plays no role).
Namely, let $N(G)\subset 2^G$ be ...
- 60.4k
11
votes
Accepted
Do acyclic amenable groups exist?
(1) Acyclic amenable groups do exist, because binate amenable groups exists: for instance, Philipp Hall's "universal locally finite group", which is by definition the Fraïssé limit of all ...
- 126
10
votes
A "simpler" description of the automorphism group of the lamplighter group
Let $R = \mathbb{Z}_N[X^{\pm 1}]$ be the Laurent polynomials ring over $\mathbb{Z}_N = \mathbb{Z}/N \mathbb{Z}$ , let $U$ be the unit group of $R$ and let $\theta$ be the ring automorphism of $R$ ...
- 7,463
10
votes
Analogous results in geometric group theory and Riemannian geometry?
Here is a very classical example. As stated in the comments, Gromov was an early proponent of importing ideas from geometry to group theory, but already thirty years earlier there was work in this ...
10
votes
Accepted
Where to find English translation of Pansu's paper from Ann. Math?
You can access the paper here:
P. Pansu, Carnot-Carathéodory metrics and quasi-isometries of rank-one symmetric spaces Ann. of Math. (2) 129 (1989), no. 1, 1–60. (English translation).
Or here.
Let me ...
- 27.3k
9
votes
Accepted
Is there a simple group that is torsion-free, type $\textrm{F}_\infty$, and infinite dimensional?
A while ago, James Hyde and Yash Lodha found a group with all four of these properties, I should mention this here. The paper is https://arxiv.org/abs/2302.04805 (to appear in Ann. Sci. Ecole Normale ...
- 3,363
9
votes
Group generated by two irrational plane rotations
A generic (i.e., outside a countable union of Zariski-closed proper subsets) $k$-tuple in $G=\mathrm{SO}(2)\ltimes\mathbf{R}^2$ freely generated a free metabelian group. In particular, it freely ...
- 60.4k
8
votes
Accepted
Question to limit groups (over free groups)
You need to prove the following folklore lemma, which is well known to researchers in the field but perhaps not written down anywhere. The proof is a nice exercise.
Folklore lemma: Let $S$ be a ...
- 24.2k
8
votes
Road map to learn about $\operatorname{Out}{F_n}$
I have course notes available on Out(F_n): https://websites.umich.edu/~alexmw/Math636Notes.pdf
- 81
8
votes
Amalgamated product acting on CAT(0) cube complex
To extend the gluing result from Bridson--Haefliger to non-positively curved cube complexes, it is important to work in the correct category.
If we want the result to also be a non-positively curved ...
- 24.2k
8
votes
Residually solvable Bianchi groups
This is true for $d=3$. Let $\zeta=\frac{1+\sqrt{-3}}{2}$, $\mathcal{O_3}=\mathbb{Z}[\zeta]$. The principal congruence subgroup of $PSL_2(\mathcal{O}_3)$ of level $1+\zeta$, which divides $3$, is a ...
- 66.8k
7
votes
An application of ping-pong lemma
As another version of HJRW’s answer: free groups are (isomorphic to) fundamental groups of graphs. The rank is then captured by the number of “independent” loops in the graph. For example $F_2$ is ...
- 26.4k
7
votes
An application of ping-pong lemma
(This is an elaboration of HJRW’s comments on the question and on Sam Nead’s answer.)
This can be seen using the representation of free groups as fundamental groups of graphs, and subgroups as ...
- 19.2k
7
votes
An application of ping-pong lemma
Fact. In a free product $A\ast B$, the groups $aBa^{-1}$ for $a\in A$ generate their free product. In other words, the homomorphism $j:B^{\ast A}\to A\ast B$ mapping $b$ in the $a$-th copy to $aba^{-1}...
- 60.4k
7
votes
Accepted
Extreme amenability of topological groups and invariant means
The action $G\curvearrowright\beta G$ is continuous iff $G$ is discrete, so for nondiscrete groups it is not true that $G$ is extremely amenable iff this action has a fixed point.
What one should look ...
- 2,904
7
votes
Accepted
Dualizing module for $\operatorname{Aut}(F_n)$
In our paper here, Himes, Miller, Nariman, and myself prove that Hatcher-Vogtmann’s question has a negative answer, at least for $n=5$. It also probably has a negative answer for larger $n$, but our ...
- 43.6k
6
votes
Reference for Chebyshev centers
The name "Chebyshev center" was introduced by Garkavi [1], for the relationship to the Cheybshev approximation problem (minimize the maximum error). Garkavi refers to a 1951 paper by ...
- 178k
6
votes
Accepted
Commensurability classes of subgroups of a nilpotent group
No. Let $U_n(R)$ be group upper triangular $n\times n$ matrices with identity diagonal over the ring $R$.
The groups $U_3(\mathbf{Z}[\sqrt{d}])$ are pairwise non-abstractly-commensurable, when $d\ge 2$...
- 60.4k
6
votes
Residually solvable Bianchi groups
(Inspired by Ian Agol's answer for d=3)
It seems to be true if $d$ is not $19$ modulo $24$. Indeed in this case $\mathcal{O}_d$ surjects as a ring onto either $\mathbf{Z}/2\mathbf{Z}$ or $\mathbf{Z}/3\...
- 60.4k
Only top scored, non community-wiki answers of a minimum length are eligible