We give a unified solution to the conjugacy problem for Thompson's groups F, T , and V . The solution uses "strand diagrams", which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson's groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson's groups, and we use this correspondence to investigate the dynamics of elements of F. Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.
We introduce forest diagrams to represent elements of Thompson's group F . These diagrams relate to a certain action of F on the real line in the same way that tree diagrams relate to the standard action of F on the unit interval. Using forest diagrams, we give a conceptually simple length formula for elements of F with respect to the {x 0 , x 1 } generating set, and we discuss the construction of minimum-length words for positive elements. Finally, we use forest diagrams and the length formula to examine the structure of the Cayley graph of F .
Abstract. We prove that Claas Röver's Thompson-Grigorchuk simple group V G has type F∞. The proof involves constructing two complexes on which V G acts: a simplicial complex analogous to the Stein complex for V , and a polysimiplical complex analogous to the Farley complex for V . We then analyze the descending links of the polysimplicial complex, using a theorem of Belk and Forrest to prove increasing connectivity.
Using a result of Kari and Ollinger, we prove that the torsion problem for elements of the Brin-Thompson group 2V is undecidable. As a result, we show that there does not exist an algorithm to determine whether an element of the rational group R of Grigorchuk, Nekrashevich, and Sushchanskii has finite order. A modification of the construction gives other undecidability results about the dynamics of the action of elements of 2V on Cantor Space. Arzhantseva, Lafont, and Minasyanin prove in 2012 that there exists a finitely presented group with solvable word problem and unsolvable torsion problem. To our knowledge, 2V furnishes the first concrete example of such a group, and gives an example of a direct undecidability result in the extended family of R. Thompson type groups.
arXiv:1405.0982v1 [math.GR] 5 May 2014Theorem 1.3. The finiteness problem for groups generated by asynchronous automata is unsolvable.Proof. This was proven in [7] for a 3-tape reversible Turing machine, and improved to a 1-tape, 2-symbol machine in [28]. Corollary 6.4. It is not decidable, given a reversible Turing machine and a finite starting configuration, whether the machine will halt.
We construct rearrangement groups for edge replacement systems, an infinite class of groups that generalize Richard Thompson's groups F , T , and V . Rearrangement groups act by piecewise-defined homeomorphisms on many self-similar topological spaces, among them the Vicsek fractal and many Julia sets. We show that every rearrangement group acts properly on a locally finite CATp0q cubical complex, and we use this action to prove that certain rearrangement groups are of type F8.
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