MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $F_2$ denote the free group of rank two and consider the group $G=\langle a,b,c \mathbin | a^2b^2c^2=1\rangle$ which is the fundamental group of the connected sum of three projective planes. Does $G$ have $\mathbb{Z} \times F_2$ as a subgroup? Thanks!

share|cite|improve this question
up vote 5 down vote accepted

The answer is `no'. No hyperbolic group contains a copy of $\mathbb{Z}^2$. To give some more details, the action of $\Gamma=\pi_1(3\mathbb{R}P^2)$ on the hyperbolic plane is free, discrete, and every element acts loxodromically. Commuting elements must have a common axis, but $\mathbb{Z}^2$ cannot act freely and discretely on the real line.

share|cite|improve this answer
Hey Henry, thanks for the answer. Does $F_2 \times \mathbb{Z}$ have $\mathbb{Z}^2$ as a subgroup? – dan Feb 11 '11 at 19:32
@dan Yes, of course, because $F_2$ has $\mathbb{Z}$ as subgroup. – Johannes Hahn Feb 11 '11 at 19:49

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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