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Results tagged with gt.geometric-topology
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user 32210
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
6
votes
Are negatively curved $2$-complexes homeomorphic to quotients of the form $\mathbb{H}^{2}/G$...
In this vague form your question has an easy negative answer: take two copies of the hyperbolic plane and identify them along a half-space, this gives you an example of a negatively curved space which …
- 3,399
2
votes
Accepted
Twisted torsions of reducible representations of knot groups
This should be true, here is a (hoperfully correct) sketch of proof: the Reidemeister torsion is a continuous function on the set of acyclic representations (as it can be computed from determinants of …
- 3,399
2
votes
Accepted
How does Siegel's Hilbert-Blumenthal fundamental domain differ from Götsky's?
I'm not familiar with the Götsky--Cohn construction but Siegel's (as explained in van der Geer's book) seems clear:
there is a "height function" $y$ on $X = \mathbb H^2 \times \mathbb H^2$ (the "di …
- 3,399
9
votes
Accepted
Equivalence of surjections from a surface group to a free group
This is true, and it is written up in lemma 2.2 of "The co-rank conjecture for 3--manifold groups" by C. Leininger and A. Reid https://arxiv.org/abs/math/0202261. They state the result in slightly dif …
- 3,399
3
votes
A question about congruence subgroups
This is true in general, by an argument similar to the one you used for the coprime case. The subgroup $\Gamma$ generated by $\Gamma(N_1) \cup \Gamma(N_2)$ contains the matrices
$$
a = \begin{pmatrix …
- 3,399
5
votes
Accepted
Manifolds with trivial mapping class group and large $H^1$?
I think that the construction in Belolipetsky--Lubotzky, Finite Groups and Hyperbolic Manifolds (https://arxiv.org/abs/math/0406607) provides a more algebraic construction of such manifolds for any $n …
- 3,399
13
votes
Accepted
Definition of cusped manifold?
Cusped manifolds are noncompact complete hyperbolic manifolds with finite Riemannian volume.
More precisely, a cusped hyperbolic n-manifold is a Riemannian manifold (without boundary) of constant ne …
- 3,399
2
votes
0
answers
61
views
Critical exponent for groups with parabolics
I'm going to ask this question first in classical setting and then sketch its natural geometric setting.
Let $\Gamma$ be a subgroup of $\operatorname{PSL}_2(\mathbb Z)$ (the question is mostly interes …
- 3,399
5
votes
How bad is the modular space?
I assume you mean for $K$ to be an imaginary quadratic field (see Misha's comment). There have been computations of the homology for a substantial array of small values of the discriminant of $K$, by …
- 3,399
2
votes
For an arithmetic hyperbolic 3-manifold group, when is its trace field not its invariant tra...
Here is a Fuchsian construction that I think works: fix a prime number $p$, and let
$$
\Gamma(p) = \left\{ g \in \mathrm{PSL}_2(\mathbb Z) :\: g = 1 \pmod{p}\right\}
$$
and let $\gamma \in \mathrm{P …
- 3,399
8
votes
Just how close can two manifolds be in the Gromov-Hausdorff distance?
This is only an answer to one point of your question: for surfaces of large genus $g$ the distance should be
$$
d(S, \mathrm{point}) \asymp \log(g).
$$
The lower bound should follow from volume esti …
- 3,399
4
votes
Quasi-isometry and left invariant orderability for groups
There are also lots of amenable examples: by a theorem of P. Linnell and D. Witte-Morris (in this paper), an amenable group is left-orderable if and only if it is locally indicable (i.e. any non-trivi …
- 3,399
4
votes
Lattices of PU(n,1) with large abelianization
There are sequences of congruence covers of certain arithmetic complex hyperbolic surfaces with unbounded first Betti number, as proven in this paper of Simon Marshall: https://arxiv.org/abs/1301.7244 …
- 3,399
2
votes
Blaschke Condition for hyperbolic lattices
I cannot make sense of the other answer and I think the sum is infinite so let me add mine.
If I understood the question correctly the tiling with Schläfli symbol $\{p, q\}$ is just the tiling of th …
- 3,399
1
vote
Relation between conjugacy class, quotient isomorphism class, and signature of Fuchsian groups
First I have two comments: I am a bit surprised that you say that (D') $\Rightarrow$ (A): I believe the action on the Poincaré half-plane sees only the image in $\mathrm{PSL}_2(\mathbb Z)$. Also, your …
- 3,399