Can the the monodromy matrix of the action of a pseudo-Anosov homeomorphism of a surface on it's homology be of the form of a reducible or block matrix? Any such reference can be helpful. Thanks.

  • $\begingroup$ "some space" is a bit unprecise... $\endgroup$ – YCor Jun 8 '17 at 10:43
  • $\begingroup$ Sorry. I meant "can it be possible"? Apologies again. $\endgroup$ – diptocal47 Jun 8 '17 at 18:28
  • $\begingroup$ I was asking about the meaning of space: vector space, topological space, etc, especially because "surface" appears in the title and not in the question. $\endgroup$ – YCor Jun 8 '17 at 21:18
  • $\begingroup$ Yes, I meant surfaces. Sorry for the lame writing. $\endgroup$ – diptocal47 Jun 9 '17 at 6:53
  • $\begingroup$ The question needs rewriting to make much sense. Let's assume the previous suggested correction: "surface" instead of "space". A surface does not have a "pseudo-Anosov monodromy matrix", but perhaps a homeomorphism of a surface does; should your question be "can the monodromy matrix of a pseudo-Anosov homemorphism of some surface be..."? If so, then what do you mean by "the monodromy matrix of a homeomorphism of a surface"? Do you mean the monodromy matrix of the action of that homeomorphism on the homology of the surface (as assumed in the answer of @IgorRivin)? $\endgroup$ – Lee Mosher Jul 8 '17 at 13:51

Every symplectic matrix arises as the monodromy of a pseudo-Anosov homeomorphism. The canonical reference is Farb-Margalit:

Farb, Benson; Margalit, Dan, A primer on mapping class groups, Princeton Mathematical Series. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/ebook). xiv, 492 p. (2011). ZBL1245.57002.

Where you can find the facts that there exists pseudo-Anosovs in the Torelli group (trivial monodromy), and the map from the mapping class group to the Symplectic group is surjective. Taking some element with your favorite matrix, and multiplying it by a high power of a Torelli pseudo-anosov will give you what you want.

  • $\begingroup$ "The canonical reference is Farb-Margalit" :) $\endgroup$ – Cusp Jun 8 '17 at 14:03
  • $\begingroup$ Thanks. I had read it previously, but was not sure overall. It is indeed a great book. Thanks again. $\endgroup$ – diptocal47 Jun 8 '17 at 18:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.