Exponential equations in groups

An exponential equation over a group G is an equation of kind $u_1g_1^{x_1}.... u_ng_n^{x_n}=1$, where $u_i$, $g_i$ are given elements of G and $x_i$ are variables with possible values in $Z$. In the joint paper with A. Bier we show that if G is acylindrically hyperbolic, then the norm of a "minimal solution" of such equation can be linearly bounded in terms of lengths of its coefficients $u_i$, $g_i$. In the joint paper with A. Iwanow we show that there exists a finitely presented group $G$ such that there is an algorithm solving exponential equations with one variable over $G$ and there is no algorithm solving exponential equations with two variables over $G$. In my talk I will sketch the proofs of these and related results.