text: abstract
  in this paper, using the notions graphs, core graphs, immersions and covering maps
  of graphs, introduced by stallings in 1983, we prove the burnside condition for the
  intersection of subgroups of free groups with burnside condition.
  keywords: graph, fundamental group, immersion and covering theory, burnside
  condition.
  2010 msc: 05e15, 05e18, 55q05, 57m10
  1. introduction and motivation
  in [3] j. stallings studied on free groups by theory of graphs. he introduced the
  concept of immersions of graphs, provided an algorithmic process to study on finitely
  generated subgroups of free groups. using these tools, he also gave an elegant proof
  for howson’s theorem ”if a and b are finitely generated subgroups of a free group,
  then a ∩ b is finitely generated”. moreover, using immersions of graphs and core
  graphs (graphs with no trees hanging on) some mathematicians such as everitt and
  gersten studied on h. neumann’s inequality on the rank of a ∩ b (see [1] and [2]).
  stallings also in [3], introduced another notation called ”burnside condition for a
  subgroup”. in this paper, we focus on this notion, and using similar methods, we
  prove that if a and b are finitely generated subgroups of a free group f , and a ∩ b
  satisfies the burnside condition in both a and b, then a ∩ b satisfies the burnside
  condition in a ∨ b, the subgroup of f generated by a ∪ b.
  corresponding author
  email addresses: h_mirebrahimi@um.ac.ir (hanieh mirebrahimi),
  fatemeh.ghanei91@gmail.com (fateme ghanei)
  ∗