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Take any finitely-presented group $G$ with undecidable word problem. Then $G$ is not quasi-isometric to any finitely generated group with decidable word problem, in particular, to any residually-finit …

Montesinos wrote several papers defining the meaning of branched coverings and proving basic properties(not just between manifolds, but for general topological spaces): Montesinos-Amilibia, José María …

The answer is positive and follows from Corollary 2 in Palais, Richard S., Extending diffeomorphisms, Proc. Am. Math. Soc. 11, 274-277 (1960). ZBL0095.16502. (A caveat: Palais is not entirely clear ab …

Let me convert my comment to an answer: Your expectation is false at least if the dimension of your manifold is $\ge 5$. Indeed, in every odd dimension $m\ge 5$ Milnor (Corollary 12.9 and Example 12.1 …

Consider a small complex 1-dimensional disk $D\subset Y$ transversal to $B$ and let $c$ denote the image in $\pi=\pi_1(Y-B)$ of the oriented loop $\partial D$. Let $n$ denote the order of the image of …

Try sections 1-15 of this paper: Cannon, James W.; Floyd, William J.; Kenyon, Richard; Parry, Walter R., Hyperbolic geometry, Levy, Silvio (ed.), Flavors of geometry. Cambridge: Cambridge University P …

Even for coarse maps between Gromov-hyperbolic spaces $f: X\to Y$ there are neither reasonable upper nor lower bounds of the type $$ \psi_-((x,y)_z)\le (f(x), f(y))_{f(z)}\le \psi_+((x,y)_z) $$ (where …

In fact, more is true and you do not need separate arguments for rank 1 and higher rank. The following is Theorem 13.1(i) in the book of Ballmann, Gromov and Schroeder "Manifolds of nonpositive curv …

Start with an Osgood curve $C$, a Jordan curve in $R^2$ of positive 2-dimensional measure. The curve $C$ bounds a domain $\Omega$ in $R^2$ diffeomorphic to $R^2$. Lastly, take the Cartesian product …

Your question (actually, questions) is a bit too vague for my taste. Here is an answer of sorts. If we replace $\mathbb{H}^k$ by a Hadamard manifold with variable (but bounded) curvature, can we em …

I do not have a self-contained reference, but the key is Long, D. D.; Reid, A. W., Constructing hyperbolic manifolds which bound geometrically, Math. Res. Lett. 8, No. 4, 443-455 (2001). ZBL0992.57023 …

While this might have been known earlier, one way to derive this result is to apply Corollary 1.2 in Edwards, R. D.; Kirby, R. C., Deformations of spaces of imbeddings, Ann. Math. (2) 93, 63-88 (1971) …

Yes, this is true, but proving this is easier than finding a reference. Every finitely-generated matrix group (e.g. a lattice in $PSL(2, {\mathbb R})$ contains a torsion-free subgroup. The general re …

As it was made quite clear in the comments, you are not at the stage where you can ask a sensible question. Thus, I am treating your question as a reference request. The first issue is that there is n …

First of all, the constants $c_i$ will have to depend on the complex structure of $X$ since without prescribing a complex structure one cannot talks about dependence on a basis of the space of holomor …