(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 2, PAGES 167-176

## Pseudocharacters on free groups, invariant with respect to some types of endomorphisms

D. Z. Kagan

Abstract

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We investigate methods for constructing nontrivial pseudocharacters on free group $F$n invariant with respect to certain types of endomorphisms. We find some conditions for endomorphisms of the free group under which there is a nontrivial pseudocharacter that is invariant with respect to these endomorphisms. We consider free products $R = \tilde R * \prod_\left\{i=k\right\}^n \langle r_i \rangle$, where one factor is $F$n, and the other factor is a group on which there is a pseudocharacter. For such products we obtain a similar result about the conditions of existence of nontrivial pseudocharacters invariant with respect to certain endomorphisms.

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