All Questions
20 questions
3
votes
0
answers
115
views
Finite homology of a homogeneous space
Let $\Gamma$ be a cocompact lattice in $\operatorname{SL}(2,\mathbb R)$ and $X=\operatorname{SL}(2,\mathbb R)/\Gamma$ be the underlying homogeneous space. Can the homology group $H_1(X,\mathbb Z)$ be ...
7
votes
1
answer
240
views
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 ...
- 1,097
7
votes
2
answers
617
views
Status of the Hopf-Thurston sign conjecture in dimension 4
A famous conjecture in topology asserts:
The Euler characteristic of a closed aspherical $2n$-manifold $M$ satisfies $(-1)^n\chi(M) \geq 0$.
This was conjectured by Hopf for manifolds with non-...
- 11.9k
6
votes
1
answer
658
views
Generalized Birman exact sequence for surfaces with boundaries
Let $S_g^n$ be a surface of genus g with n boundaries and let $Mod(S_g^n)$ be its mapping class group.
We will also denote by $S_{g,m}^n$ a surface of genus g with n boundaries and m punctures.
The ...
5
votes
1
answer
336
views
"Simplicial complex" product of groups?
Let $X=(V,E)$ be a graph, and to each vertex $v \in V$, associate a group $G_v$. The graph product of the groups $G_v$ (as defined e.g. here) is $F/R$; the quotient of the free product of the $G_v$ by ...
- 208
13
votes
1
answer
289
views
Powers of the Euler class, torsion free subgroup of Homeo($S^1$)
For any subgroup $G$ of $\text{Homeo}(S^1)$, we have the Euler class $\chi$ in the group cohomology $H^2(G;\mathbb{Z})$. One can think of this class as the pullback of the generator of $H^2(\mathrm{B}\...
- 1,003
12
votes
1
answer
309
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 ...
- 965
5
votes
1
answer
280
views
Curvature and asphericity of cube complexes
Let $K$ be a connected cube complex (one may assume that its a cellulation of a smooth, closed manifold). Such a $K$ comes equipped with a length metric (one assumes that each edge is of unit length). ...
- 1,017
26
votes
1
answer
615
views
What is the minimal dimension of a complex realising a group representation?
This question is inspired by this one, which was about representations that can be realised homologically by an action on a graph (i.e., a 1-dimensional complex).
Many interesting integral ...
- 10.9k
3
votes
1
answer
267
views
In what sense is every element of $H_2(G)$ "represented by a free action on some surface"
(This is a cross-post of this unanswered math.stackexchange question)
In Edmond's 1982 paper Surface Symmetry II, at the bottom of page 145, he writes:
"Corollary - If $G$ is a split nonabelian ...
- 7,527
3
votes
1
answer
432
views
Relation between conjugacy class, quotient isomorphism class, and signature of Fuchsian groups
Let $\Gamma\le SL(2,\mathbb{Z})$ be a finite index subgroup, not necessarily "congruence".
Let $c_4,c_6$ be the number of conjugacy classes of elements of order 4 and 6 respectively, let $c_{-1}$ be ...
- 7,527
10
votes
4
answers
578
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}) = ...
- 101
1
vote
0
answers
278
views
Homology of spherical braid groups
By the spherical braid group, I mean the fundamental group of the configuration space of distinct unordered points in $S^2$. I am wondering what is known about the group homology of the spherical ...
- 116
11
votes
1
answer
337
views
Growth of Poincaré duality groups
Can one prove that Poincaré duality groups cannot have intermediate growth?
- 4,188
10
votes
2
answers
890
views
Are virtual cubulated groups cubulated?
Suppose $G$ has a finite index subgroup $N$ such that $N$ acts properly and cocompactly on a CAT(0)-cube complex. Does $G$ also act properly and cocompactly on a CAT(0)-cube complex?
Edit: After ...
- 539
8
votes
1
answer
692
views
Classification of geometric outer automorphisms of free groups
Good evening everyone,
an outer automorphism $[\phi]\in Out(F_n)$ is geometric if it is induced by a surface homeomorphism $h\colon M\stackrel{\cong}{\to}M$, where $M$ is a compact surface with ...
- 937
17
votes
3
answers
1k
views
The second homotopy group of a simple CW-complex
Let $X$ be a CW-complex with
one 0-cell
two 1-cells
three 2-cells
no cells in dimensions 3 or higher.
Is it always true that $\pi_2(X)\ne 1$?
- 1,181
10
votes
0
answers
458
views
is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?
This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...
- 101
18
votes
2
answers
790
views
The kernel of the map from the handlebody group to Outer automorphisms of a free group
Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the ...
- 6,229
5
votes
2
answers
666
views
HNN extensions which are free products
which HNN-extensions are free products? this question is related with another still unsolved about Nielsen-Thruston-reducibility and connected-sum-irreducibility of 3d-torus- bundles...
- 345