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Riffing off of Ben Steinberg's answer, which amounts to the statement that a tree is not quasi-isometric to the hyperbolic plane, here's a proof which doesn't require knowing anything about ends. Til …

Here's an example. Start with a $4g$-gon $P$ with opposite sides glued, which yields a closed surface of genus $g$. Between any side-pair which is glued, there are $2g-1$ sides, which is the maximum p …

Assuming that you DO mean that $T^n$ is a finite sheeted covering space, at the very least one can say that $\pi_1(M)$ is a torsion free $n$-dimensional crystallographic group. This follows from the B …

In my proof that mapping class groups are automatic, Ann. of Math. (2) 142 (1995), no. 2, 303–384, I used a theorem from ECHLPT "Word Processing in Groups" which says that if a groupoid is automatic …

You can build a certain covering space of the surface $S$ rather explicitly as a nested union of closed discs $D_1 \subset D_2 \subset D_3 \subset \cdots$, each contained in the interior of the next, …

One way this formula is proved is using intersection number of oriented curves. The proof works just fine on any orientable surface using coefficients in any field, both for defining Betti numbers and …

I remember this chestnut. Taken together, your two equations say that the torus $i(T)$ is contained in a 3-ball $B$ embedded in $S^2 \times S^1$: the first equation says that $i(T)$ is obtained up to …

To understand the difficulties inherent in forming a "path-based" definition of the orbifold fundamental group, it is good to ponder Serre's definition of a fundamental group of a graph of groups, giv …

The machinery of Markov partitions and stable/unstable foliations for Anosov and Axiom A diffeomorphisms was adapted to several different bits of low dimensional topology. In the Nielsen-Thurston clas …

If you are willing to quote the Schoenflies theorem and the classification of surfaces, a quick proof of this result is a standard exercise. First, using algebraic topology (as in the proof of the Jor …

I think the short answer is "No", you cannot deduce from this result a classification of all geometric outer automorphisms. I think it might eventually be possible to obtain a classification of geom …

In the topological category the proof that connected sum is well-defined depends on the Annulus Theorem, first proved by Kirby; the necessity of the Annulus Theorem is seen from Bruno Martelli's answe …

Here is an answer involving finitely generated groups in the quotient, and free groups in the kernel and the total group. Suppose that $G$ is an infinite, finitely generated group. Let $F_n = \langl …

I doubt there is a "classification", but there are some interesting examples. Two which come to mind: Harer's description of the moduli space of a Riemann surface with spin structure; and Torelli spac …

The theories for an open surface $S$ and for a compact surface with boundary $\overline S$ whose interior is identified with $S$ are the same. The inclusion of $S$ into $\overline S$ defines an isomor …