All Questions
19 questions
20
votes
2
answers
824
views
Can a finite group action by homeomorphisms of a three-manifold be approximated by a smooth action?
Let $M^3$ be a smooth three-manifold, and let $\gamma:G\to\operatorname{Homeo}(M)$ be a finite group action on $M$ by homeomorphisms.
Can $\gamma$ can be $C^0$-approximated by smooth group actions $...
- 18.7k
12
votes
2
answers
691
views
Is the fundamental group of any compact hyperbolic 3-manifold embeddable into a p-adic group?
Is it true that for every compact hyperbolic $3$-manifold $M$ there exists a prime $p$, a finite field extension $K/\mathbb{Q}_p$, and an injective group homomorphism $$\tau \colon \pi_1(M) \to \...
- 11.3k
10
votes
2
answers
460
views
Presentations of mapping class groups in dimension $3$
For any closed oriented surface $M$, its mapping class group $MCG(M)$ can be generated by Dehn twists along certain curves on $M$. A presentation for the group $MCG(M)$ was found in [1] and then ...
- 5,230
9
votes
2
answers
886
views
Hyperbolic $3$-manifold groups that embed in compact Lie groups
Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group?
It is known that every surface group can be embedded into any semisimple ...
- 29.1k
9
votes
3
answers
735
views
Judging whether a finitely presented group is a 3-manifold group?
Given a finitely presented group $G$, how many necessary conditions do people know for $G$ to be isomorphic to the fundamental group of some closed connected 3-manifold? (e.g. residually finite)
8
votes
2
answers
489
views
Quantitative word problem for 3-manifold groups
The word problem for 3-manifold groups is solvable: given a based loop $\gamma$ in $M^3$ there is an algorithm to decide whether $\gamma$ bounds a disk.
What kinds of quantitative results are known ...
- 595
8
votes
1
answer
387
views
Outer automorphism group of Brieskorn homology sphere?
In this post, it is discussed how a Brieskorn homology sphere $\Sigma(a_1,a_2,a_3)$ with $\displaystyle \frac{1}{a_1}+ \frac{1}{a_2}+ \frac{1}{a_3} < 1$ is an aspherical manifold with a ...
- 309
7
votes
1
answer
358
views
Virtually large groups of small rank (related to 3-manifolds)
Edited 25.05.21: the assumptions of the question were incorrect, but as the discussion may be helpful for future MOnauts, I'll strike my mistakes and add clearly marked explanations afterwards.
I am ...
- 323
7
votes
1
answer
372
views
Two details from Stallings's proof of the sphere theorem
EDIT: After a little prompting by Mark Grant, I answered the first question in the comments. The second question remains open.
Let $M$ be a compact $3$-manifold with $\pi_2(M) \neq 0$. The sphere ...
- 353
5
votes
2
answers
456
views
Is the mapping torus of an automorphism of a free group virtually an amalgamated product?
Let $F$ be a nonabelian finitely generated free group,
let $\tau \in \mathrm{Aut}(F)$ be an element of infinite order,
and set $G = F \rtimes \mathbb{Z}$,
where the action of $\mathbb{Z}$ on $F$ is ...
- 11.3k
5
votes
0
answers
96
views
$M^3$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not virtually nilpotent?
Let $M$ be a closed, orientable, irreducible 3-manifold and having an infinite fundamental group. Is it true that $M$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not ...
- 605
4
votes
2
answers
286
views
Geometrisation of inclusion-like epimorphisms to free groups
Let $H_g$ be the standard $3$-dimensional handle-body, whose boundary is denoted $S_g$, the oriented closed surface of genus $g\geq 1$.
Call $F_g$ be the free group of rank $g$.
Denote by $i:S_g \to ...
- 43
4
votes
1
answer
242
views
Which 3-manifolds have positive rank gradient?
For which $3$-manifolds $M$ is the fundamental group $\pi_1(M)$
finitely generated and has positive rank gradient?
Recall that the rank gradient of a finitely generated group $G$ is defined to be $$...
- 11.3k
4
votes
1
answer
192
views
Are fundamental groups of web complements residually finite?
While thinking of whether any web (spatial trivalent graph) without an embedded bridge can be realized as a branching locus of a finite branched cover over $S^3$, I realized that this problem is ...
- 1,430
4
votes
0
answers
200
views
3-manifold proof of Grushko's theorem
Grushko's theorem says that given an epimorphism $\phi: F \to G_1 * G_2$ where $F$ is a finitely-generated free group, there exists. subgroups $F_1$ and $F_2$ of $F$ so that $F = F_1 * F_2$ and $\phi(...
- 5,349
3
votes
2
answers
448
views
Quasi-isometry and left invariant orderability for groups
Is the property of left invariant orderability for finitely generated groups preserved by quasi-isometrics? More precisely, if $G$ is a left orderable (finitely generated) group and $H$ is a torsion-...
3
votes
1
answer
160
views
Does the fundamental group of a compact 3-manifold induce full profinite topology on the fundamental group of its boundary?
Let $M^3$ be a compact, orientable, irreducible 3-manifold with incompressible boundary. Let $S\subseteq\partial M$ be one of its boundary components. Does $\pi_1(M)$ induce the full profinite ...
- 605
2
votes
1
answer
257
views
Engulfing Kleinian groups?
Let $G$ be a Kleinian group, and let $H \lneq G$ be a finitely
generated subgroup. Must there be a proper finite index subgroup $U$ of $G$ containing $H$ ?
I know that this is true for Fuchsian ...
- 11.3k
1
vote
1
answer
154
views
On the realization of a compact surface as a leaf of an analytic foliation
Let $S$ be a compact orientable surface of genus $g \geq 2$. Is there any transversely real (or complex) analytic codimension one foliation $\mathcal{F}$ such that $\mathcal{F}$ has $S$ as a leaf with ...