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The answer is 'yes' by Britton's lemma (see wikipedia and, more generally, Serre's book Trees and Scott and Wall's article 'Topological methods in group theory'). Since $M$ and $U$ are smooth and c …

I think this is a great question, as there is still a need for an authoritative reference about (word-)hyperbolic groups. Since the textbook doesn't exist, I'd like to take the question in a slightly …

Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But it' …

Many examples can be exhibited using a theorem of Gersten: Theorem (Gersten): Let $G$ be a hyperbolic group of cohomological dimension 2. Every finitely presented subgroup $H$ of $G$ is hyperbolic. Th …

Definition: Let $R$ be a commutative ring with 1. Endow the power set $2^R$ with the product topology. The ideal space $\mathcal{I}(R)$ is defined to be subset of $2^R$ consisting of ideals, equippe …

It's a famous open question whether every word-hyperbolic group is residually finite. Kapovich--Wise showed that this is equivalent to asking whether every non-trivial word-hyperbolic group has non-t …

This was proved by Jack Button and Robert Kropholler: arXiv link, see p.27. (Added: But there's a caveat; see the second update below.) J.O. Button, R. Kropholler Nonhyperbolic free-by-cyclic and on …

Let $F$ be a finitely generated free group. Question: Is there an authoritative survey of analogues of the curve complex for $\mathrm{Out}(F)$? If not, as seems likely, would a passing expert be …

Many more examples, including ones where the free kernel is finitely generated, arise by looking at knot complements. Let $K$ be any non-trivial knot with Alexander polynomial $\Delta_K(t)=1$, (appare …

Let $C$ be the compact cylinder $S^1\times [0,1]$. A 3-manifold $M$ with incompressible boundary is called acylindrical if every map $(C,\partial C)\to (M,\partial M)$ that sends the components of $\ …

Perhaps it's worth mentioning that Property (T) seems to repel certain strengthenings of residual finiteness. An open question of Long and Reid asks: Is there an infinite finitely generated group …

There are many further examples with local cut points, which can be obtained by amalgams over $\mathbb{Z}$, as @YCor suggests in his comment. Perhaps the easiest example is obtained by gluing three on …

I'm late to the party, but here's how I think about this fact. The approach I will present is perhaps slightly higher tech than some, but has the advantage that it's a completely general way of comput …

In comments, the OP indicates that what they really want is a uniqueness result for reduced words in arbitrary graphs of groups. (Indeed, what the question actually asks for, that any word can be tran …

For infinite discrete groups: Lyndon & Schupp is authoritative for classical, combinatorial methods. Bridson & Haefliger has a lot of material for more geometric classes, like hyperbolic and CAT(0) …