Post's correspondence problem for hyperbolic and virtually nilpotent groups

Post's Correspondence Problem (the PCP) is a classical decision problem in theoretical computer science that asks whether for pairs of free monoid morphisms $g, h\colon\Sigma^*\to\Delta^*$ there exists any non-trivial $x\in\Sigma^*$ such that $g(x)=h(x)$. Post's Correspondence Problem for a group $\Gamma$ takes pairs of group homomorphisms $g, h\colon F(\Sigma)\to \Gamma$ instead, and similarly asks whether there exists an $x$ such that $g(x)=h(x)$ holds for non-elementary reasons. The restrictions imposed on $x$ in order to get non-elementary solutions lead to several interpretations of the problem; we consider the natural restriction asking that $x \notin \ker(g) \cap \ker(h)$ and prove that the resulting interpretation of the PCP is undecidable for arbitrary hyperbolic $\Gamma$, but decidable when $\Gamma$ is virtually nilpotent. We also study this problem for group constructions such as subgroups, direct products and finite extensions. This problem is equivalent to an interpretation due to Myasnikov, Nikolev and Ushakov when one map is injective.Comment: 17 page

Compressed Decision Problems in Hyperbolic Groups

We prove that the compressed word problem and the compressed simultaneous conjugacy problem are solvable in polynomial time in hyperbolic groups. In such problems, group elements are input as words defined by straight-line programs defined over a finite generating set for the group. We prove also that, for any infinite hyperbolic group G, the compressed knapsack problem in G is NP-complete

The Complexity of Knapsack Problems in Wreath Products

We prove new complexity results for computational problems in certain wreath products of groups and (as an application) for free solvable group. For a finitely generated group we study the so-called power word problem (does a given expression $u_1^{k_1} \ldots u_d^{k_d}$, where $u_1, \ldots, u_d$ are words over the group generators and $k_1, \ldots, k_d$ are binary encoded integers, evaluate to the group identity?) and knapsack problem (does a given equation $u_1^{x_1} \ldots u_d^{x_d} = v$, where $u_1, \ldots, u_d,v$ are words over the group generators and $x_1,\ldots,x_d$ are variables, has a solution in the natural numbers). We prove that the power word problem for wreath products of the form $G \wr \mathbb{Z}$ with $G$ nilpotent and iterated wreath products of free abelian groups belongs to $\mathsf{TC}^0$. As an application of the latter, the power word problem for free solvable groups is in $\mathsf{TC}^0$. On the other hand we show that for wreath products $G \wr \mathbb{Z}$, where $G$ is a so called uniformly strongly efficiently non-solvable group (which form a large subclass of non-solvable groups), the power word problem is $\mathsf{coNP}$-hard. For the knapsack problem we show $\mathsf{NP}$-completeness for iterated wreath products of free abelian groups and hence free solvable groups. Moreover, the knapsack problem for every wreath product $G \wr \mathbb{Z}$, where $G$ is uniformly efficiently non-solvable, is $\Sigma^2_p$-hard

A Characterization of Wreath Products Where Knapsack Is Decidable

The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and Ushakov. It is defined for each finitely generated group $G$ and takes as input group elements $g_1,\ldots,g_n,g\in G$ and asks whether there are $x_1,\ldots,x_n\ge 0$ with $g_1^{x_1}\cdots g_n^{x_n}=g$. We study the knapsack problem for wreath products $G\wr H$ of groups $G$ and $H$. Our main result is a characterization of those wreath products $G\wr H$ for which the knapsack problem is decidable. The characterization is in terms of decidability properties of the indiviual factors $G$ and $H$. To this end, we introduce two decision problems, the intersection knapsack problem and its restriction, the positive intersection knapsack problem. Moreover, we apply our main result to $H_3(\mathbb{Z})$, the discrete Heisenberg group, and to Baumslag-Solitar groups $\mathsf{BS}(1,q)$ for $q\ge 1$. First, we show that the knapsack problem is undecidable for $G\wr H_3(\mathbb{Z})$ for any $G\ne 1$. This implies that for $G\ne 1$ and for infinite and virtually nilpotent groups $H$, the knapsack problem for $G\wr H$ is decidable if and only if $H$ is virtually abelian and solvability of systems of exponent equations is decidable for $G$. Second, we show that the knapsack problem is decidable for $G\wr\mathsf{BS}(1,q)$ if and only if solvability of systems of exponent equations is decidable for $G$

Exponent equations in HNN-extensions

We consider exponent equations in finitely generated groups. These are equations, where the variables appear as exponents of group elements and take values from the natural numbers. Solvability of such (systems of) equations has been intensively studied for various classes of groups in recent years. In many cases, it turns out that the set of all solutions on an exponent equation is a semilinear set that can be constructed effectively. Such groups are called knapsack semilinear. Examples of knapsack semilinear groups are hyperbolic groups, virtually special groups, co-context-free groups and free solvable groups. Moreover, knapsack semilinearity is preserved by many group theoretic constructions, e.g., finite extensions, graph products, wreath products, amalgamated free products with finite amalgamated subgroups, and HNN-extensions with finite associated subgroups. On the other hand, arbitrary HNN-extensions do not preserve knapsack semilinearity. In this paper, we consider the knapsack semilinearity of HNN-extensions, where the stable letter $t$ acts trivially by conjugation on the associated subgroup $A$ of the base group $G$. We show that under some additional technical conditions, knapsack semilinearity transfers from base group $G$ to the HNN-extension $H$. These additional technical conditions are satisfied in many cases, e.g., when $A$ is a centralizer in $G$ or $A$ is a quasiconvex subgroup of the hyperbolic group $G$.Comment: A short version appeared in Proceedings of ISSAC 202