Abstract 2
1. Introduction 2
1.1. Dead-ends and lifting of geodesic words 3
1.2. Normal form structure 3
2. Integer normal form 4
2.1. Polycyclic groups 6
2.2. Free Nilpotent groups 7
References 14
1. Introduction
We use basic terminology and notions of geometric group theory. The central notions of the present thesis are geodesic words, isoperemetric problems and normal form of an elements in groups. In geometric group theory, for a particular group with a given finite generating set one can study two main problems:
• Description of geodesic paths (Geo)
• Description of solutions to isoperemetric problem (Isop)
Relatively comprehensible solutions of these two problems for a given group with fixed generating set usually implies high level of geometric intuition and important basis for any further studies. Even though Geo and Isop are classical motives in geometric group theory, unfortunately, there are no common techniches or methods for investigations into these problems. There exists well-developed theory which implies strong connection between algebraic and geometric properties of groups with solved Geo and Isop. This connection are based on studying of growth functions, geodesic growth function, and Dehn’s area function. For the sake of formalism, let’s precisely define what the solution of Geo and Isop is:
Definition 1.1. Let G = (S | R) be a finitely generated group with a given presentation. Solution of Geo for G with respect to S is the following set with non-trivial criteria of membership to the set:
LS(G) = {w e F(S) | w - geodesic word G}
Definition 1.2. Let G = (S | r-1, r2,..., rm) be a finitely presented group with a given presentation. For every word w =G 1 and fixed representation in ((R))F(S) one can consider area vector as a vector where i-th coordinate equals to signed occurrence of r in w: v(w) = (expn (w), expr2 (w),..., exprm (w)).
Definition 1.3. Solution of Isop for G with respect to (S, R) of area vector v is the following set with non-trivial criteria of membership to the set:
I(v)(S,R) = {W 6 (
Complete solution of Isop for G with respect (S, R) is the following set:
Is (G) = Ц I(v)(s.R)
veZm,v^0
Those definitions has some element of ambiguity because of phrase ”non-trivial criteria”. We hope, that this phrasing will be interpreted under some rational and aesthetic boundaries. If group and presentation is known from context, we abuse notation and drop excessive symbols.
1.1. Dead-ends and lifting of geodesic words. Combinatorial and geometric connection between Geo and Isop can be expressed via lifting of geodesic words and symmetries of deadends in cases of groups with normal form structure and groups with well-known solutions of Isop.
Definition 1.4. Let G, H be groups generated by same set S. If there is ф : G ^ H, such that ф($) = S and for every w 6 Is(H) there exists (only one!) v 6 LS(G) such that ф(и) = w, then we say that solution of Geo for H lifts to solution of Geo for G.
Definition 1.5. Let G = (S | R) be finitely presented group with a given finite generating set. Element represented by geodesic word w 6 F(S) is called dead-end with respect to S, if for any s 6 S и S-1 word ws is not geodesic. Word w itself is called dead-end too. Set of all dead-end words in G with respect to S is denoted as A(G, S).
These specific notions are motivated, where one can find deep connections between solutions of Isop or Geo and A(G,S). Using new notions, we want to demonstrate that solutions of Isop in one group become special part of solution of Geo in another group. One of the aims of current work is to algebraically predict such relation of solutions, which is presented in Section 2.
Also, lifting of geodesics imitate idea of lifting close geodesics to universal covering of a given good space. It’s important to note, that when lifting of geodesics occurs between H and G, then there is covering Cay(G) ^ Cay(H) and universal covering is always Cay(F(S)). In this sense, ’’lifting of geodesics” slightly lose its practical importance, because it doesn’t depart interesting closed curves from any arbitrary closed curves.
In this paper, we considered constructions that make it possible to study geodesic words in groups with a pleasant generating set. By a pleasant generating set is meant a structure in a group that makes it possible to think of each element as a product of generating elements of a fixed form. This structure is called the normal form. It was shown in the work that any polycyclic group has a group covering it, in which geodesic words are arranged in a predictable way, moreover, in some cases (free nilpotent groups), this group carries with it the structure of a representation that is calculated by purely geometric methods.
