Thank you ]]>

http://blog.jgc.org/2010/01/more-fun-with-toys-ikea-lillabo-train.html

]]>How does having more track piece types change things? It appears to not fundamentally change anything other than make the problem more messy.

]]>best,

Danny

]]>the two track you have mention at the beggining are almosl all you would like to know about that group.

BTW. you should have cusps (switches?) in your tracks. R^2L^{-2}R^2L^{-2} is a nice closed track!

]]>0. I think that declaring groups by generators and relations is some misidea. I think that we should investigate more groups defined by action (of generators). Alas, using presentations as tools.

1. The group generated by L and S is virtually abelian, so the word problem is easy to solve. I did not derive the representattion of it, but it is a finite index subgroup of the group generated by a rotation O by π/4 and a translation T. Then R=OT, L=TO^{-1}. (T is the vector between the ends of the track rotated by π/8.)

The relations in the group generated by T and O are: O^8, (O^4T)^2, [O^iTO^{-1},T]. Then if you want to know if a LR-word closes you write it in terms of O and T, you group conjugates of T, commute (is it the right verb for moving commuting elements?) them

to cancel (or not). There are only few left if and only if you could close the track if few steps.

2. What is the index of Γ_0:= in Γ:=? What is the presentation of Γ_0? I know tat I need to take representatives of cosets of Γ_0 together with L and R and play with Tietze moves…

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