Type of Document Dissertation Author Darbinyan, Arman Author's Email Address arman.darbinyan@vanderbilt.edu URN etd-07112018-142606 Title The word and conjugacy problems in lacunary hyperbolic groups Degree PhD Department Mathematics Advisory Committee

Advisor Name Title Alexander Olshanskii Committee Chair Denis Osin Committee Member Mark Sapir Committee Member Michael Mihalik Committee Member Thomas Kephart Committee Member Keywords

- word problem
- conjugacy problem
- lacunary hyperbolic groups
Date of Defense 2018-04-05 Availability unrestricted AbstractWe study the word and conjugacy problems in lacunary hyperbolic groups (briefly, LHG). In particular, we describe a necessary and sufficient condition for decidability of the word problem in LHG. Then, based on the graded small-cancellation theory of Olshanskii, we develop a general framework which allows us to construct lacunary hyperbolic groups with word and conjugacy problems highly controllable and flexible both in terms of computability and computational complexity.

As an application, we show that for any recursively enumerable subset $mathcal{L} subseteq mathcal{A}^*$, where $mathcal{A}^*$ is the set of words over arbitrarily chosen non-empty finite alphabet $mathcal{A}$, there exists a lacunary hyperbolic group $G_{mathcal{L}}$ such that the membership problem for $ mathcal{L}$ is `almost' linear time equivalent to the conjugacy problem in $G_{mathcal{L}}$. Moreover, for the mentioned group the word and individual conjugacy problems are decidable in `almost' linear time.

Another application is the construction of a lacunary hyperbolic group with `almost' linear time word problem and with all the individual conjugacy problems being undecidable except the word problem.

As yet another application of the developed framework, we construct infinite verbally complete groups and torsion free Tarski monsters, i.e. infinite torsion-free groups all of whose proper subgroups are cyclic, with `almost' linear time word and polynomial time conjugacy problems. %En route, we also show that every torsion free, non-elementary hyperbolic group $G$ has an infinite verbally complete quotient $check{G}$ in which each equation of the form $w=g$, where $w in F(y_1, y_2, ldots)setminus {1}$ and $gin check{G}$, has a solution that can be found algorithmically.

These groups are constructed as quotients of arbitrarily given non-elementary torsion-free hyperbolic groups and are lacunary hyperbolic.

Finally, as a consequence of the main results, we answer a few open questions.

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