A number of papers (with A. S. Rapinchuk and V. I. Chernousov) were devoted to an investigation of representation varieties $R_n(\Gamma)$ and character varieties $X_n(\Gamma)$ for some classes of finitely generated groups, in particular, for fundamental groups of compact surfaces. It is proved that if $\Gamma=\Delta_g$ is a fundamental group of a compact orientable surface of genus $g$, then $R_n(\Delta_g)$ is an irreducible $\mathbb{Q}$-rational variety for all $n$ and $g$. It is obtained a full description of representation and character varieties of fundamental groups $\G_g$ of compact non-orentable surfaces. Developed methods were used for description of $n$-dimensional representation and character varieties both of wide class of groups with one relation and several groups of $F$-type. The problem of decomposing finitely generated groups into non trivial free product with amalgamation is investigated in a number of papers. It is proved that a group with two generators and one relation with torsion is a non trivial free product with amalgamation. Decomposing generalized triangle groups is investigated. As a corollary we proved that Fuchsian groups $H_1=\langle a,b\mid [a,b]^n=1\rangle$ and $H_2=\langle a,b\mid a^2=[a,b]^n=1\rangle$, where $n\ge2$, are non trivial free products with amalgamation. It is proved that if the dimension of a character variety of representations of a finitely generated group $\Gamma$ into $SL_2(\mathbb{C})$ is greater than 1 then $\Gamma$ is a non trivial free product with amalgamation.

Graduated fromFaculty of Mathematics and Mechanics of Byelorussian State University in 1983 (department of algebra). Ph.D. thesis was defended in 1989. D.Sci. thesis was defended in 2001. A list of my work contains more than 40 titles.

• Belarusian State University, Minsk
• Institute of Mathematics, National Academy of Sciences of the Republic of Belarus, Minsk