15
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) ...

12
votes

### 4-manifold $M$ with intersection form of Leech lattice

If you're assuming $M$ is simply-connected, then it would be spin (since the Leech lattice is even). So a smooth manifold would violate Rokhlin's theorem. In the topological case (still assuming ...

9
votes

Accepted

### Abstract simplicial complexes - Reference for an elementary definition of mapping degree for simplicial maps?

I think the right generality to restrict to is the following:
Let $n$ be a positive integer, and let $X$ and $Y$ be $n$-dimensional oriented simplicial complexes, with the following properties:
Every ...

6
votes

### 0-surgery on a fibered hyperbolic ribbon knot

I had a quick look at Maggie Miller's thesis. It appears that the pretzel knots $(\pm 2, n, -n)$ where $n$ is odd are fibered ribbon knots (and they are also hyperbolic). Moreover, it appears that the ...

5
votes

### Finding a hyperbolic metric with geodesic boundary on a given Riemann surface

See Theorem 1(b) in the following paper; it is enough that the boundary is smooth.
Osgood, B.; Phillips, R.; Sarnak, P. Extremals of determinants of Laplacians. J. Funct. Anal. 80 (1988), no. 1, 148--...

5
votes

### Knot theory in handlebodies of arbitrary genus

You example of graphs in the plane "becoming embedded via local operations" is not modelled by handle attachment. (For example, if you immerse $K_5$ in a disk $D^2$ and then attach one-...

5
votes

### Knot theory in handlebodies of arbitrary genus

If you mean adding 1 and 2 handles to the boundary of the ball, that will never change the (non)triviality of the knot in the 3-ball. On the other hand, perhaps the notion of tunnel number is what you ...

5
votes

### Does every triangulable manifold have a vertex-transitive triangulation?

This is to supplement Ian's answer and get examples in all dimensions $\ge 3$.
Let $M={\mathbb H}^n/\Gamma$ be a compact hyperbolic $n$-manifold; suppose that $f\in Homeo(M)$ is a homeomorphism of ...

5
votes

### Does every triangulable manifold have a vertex-transitive triangulation?

There exists many closed connected hyperbolic 3-manifolds $M$ with trivial symmetry group, and hence trivial mapping class group. $M$ cannot be homeomorphic to a simplicial complex $\tau$ which admits ...

5
votes

Accepted

### Does $\overline{\mathbb{C}P^2 \setminus B^4}$ embed (smoothly/topologically) in $\mathbb R^5$?

$\def\RR{\mathbb{R}}\def\CC{\mathbb{C}}\def\PP{\mathbb{P}}$I think the answer is no. Topology isn't my strength, so check this argument.
Let $\infty$ be the point $[0:0:1]$ in $\CC \PP^2$ and put $U = ...

4
votes

Accepted

### Finding a hyperbolic metric with geodesic boundary on a given Riemann surface

A good reference is
W. Abikoff, The real analytic theory of Teichmuller space, Springer, 1980. (Chap. II section 1).
The idea is that you construct the double: it is the result of gluing of your ...

1
vote

Accepted

### Bring's curve $\sum_{i=1}^5 x_i^k = 0$ for $k = 1,2,3$ and an analogue $\sum_{i=1}^6 y_i^k = 0$ for $k = 1,2,4,7$

The equations $\sum_{i=1}^6 y_i^k = 0$ for $k=1,2,4,7$ do not cut out a curve because the $k=1,2,4$ equations imply the one for $k=7$. (The 7th power sum is a polynomial in the power sums of degree 1 ...

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