Where are groups of intermediate growth?

Transcription

1 Where are groups of intermediate growth? Olga Macedońska Silesian University of Technology (Poland) Będlewo, June, 2015

2 Where are groups of intermediate growth? Olga Macedońska, Silesian University of Technology (Poland)

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7 G R O U P L A N D

8 Definitions The notion of growth for groups is defined for finitely generated groups. It does not depend on the choice of a generating set S.

9 Definitions The notion of growth for groups is defined for finitely generated groups. It does not depend on the choice of a generating set S. Let g (G,S) (n) denote the number of distinct elements in G that can be expressed as products of at most n elements from S S 1.

10 Definitions The notion of growth for groups is defined for finitely generated groups. It does not depend on the choice of a generating set S. Let g (G,S) (n) denote the number of distinct elements in G that can be expressed as products of at most n elements from S S 1. The group G is of polynomial growth if there exist real numbers d, C such that n N g (G,S) (n) Cn d.

11 Definitions The notion of growth for groups is defined for finitely generated groups. It does not depend on the choice of a generating set S. Let g (G,S) (n) denote the number of distinct elements in G that can be expressed as products of at most n elements from S S 1. The group G is of polynomial growth if there exist real numbers d, C such that n N g (G,S) (n) Cn d. The group G is of exponential growth if there exists a real number C > 1 such that n N g (G,S) (n) C n.

12 From the results of Gromov (1981) and Wolf (1968, Thm. 3.2) it follows that a finitely generated group G is of polynomial growth if and only if it is nilpotent-by-finite. That is G contains a nilpotent normal subgroup of finite index.

13 From the results of Gromov (1981) and Wolf (1968, Thm. 3.2) it follows that a finitely generated group G is of polynomial growth if and only if it is nilpotent-by-finite. That is G contains a nilpotent normal subgroup of finite index.

14 Milnor and Wolf in 1968 proved: Every finitely generated soluble-by-finite group has either polynomial or exponential growth.

15 Milnor and Wolf in 1968 proved: Every finitely generated soluble-by-finite group has either polynomial or exponential growth. Milnor, Power et al. in 1968 formulated the famous question whether this dichotomy holds for all groups.

16 Milnor and Wolf in 1968 proved: Every finitely generated soluble-by-finite group has either polynomial or exponential growth. Milnor, Power et al. in 1968 formulated the famous question whether this dichotomy holds for all groups. Tits in 1972 proved:the dichotomy is true for linear groups.

17 Milnor and Wolf in 1968 proved: Every finitely generated soluble-by-finite group has either polynomial or exponential growth. Milnor, Power et al. in 1968 formulated the famous question whether this dichotomy holds for all groups. Tits in 1972 proved:the dichotomy is true for linear groups. In 1983 R. Grigorchuk gave the negative answer. He constructed examples of groups whose growth is neither polynomial nor exponential. Such groups are called now groups of intermediate growth.

18 Groups of intermediate growth were constructed by many authors, e.g. S.V. Aleshin in 1971, V.I.Sushchanskii in 1979 constructed such groups in connection with the unrestricted Burnside problem. All of these groups were residually finite.

19 Groups of intermediate growth were constructed by many authors, e.g. S.V. Aleshin in 1971, V.I.Sushchanskii in 1979 constructed such groups in connection with the unrestricted Burnside problem. All of these groups were residually finite. In 1989 R.Grigorchuk conjectured that every group of intermediate growth is residually finite.

20 Groups of intermediate growth were constructed by many authors, e.g. S.V. Aleshin in 1971, V.I.Sushchanskii in 1979 constructed such groups in connection with the unrestricted Burnside problem. All of these groups were residually finite. In 1989 R.Grigorchuk conjectured that every group of intermediate growth is residually finite.

21 The first counterexaple was constructed by A.Erschler (2004). The groups of intermediate growth constructed by A. Erschler are not residually finite but they have the residually finite quotient G first, the first Grigorchuk group.

22 The first counterexaple was constructed by A.Erschler (2004). The groups of intermediate growth constructed by A. Erschler are not residually finite but they have the residually finite quotient G first, the first Grigorchuk group.

23 The problem arrives whether a group of intermediate growth can satisfy a non-trivial law. This problem can be found in the article of A. Mann and Yves de Cornulier in the form: Can a finitely generated group satisfying a non-trivial law have intermediate growth?

24 We can prove: Theorem 1

25 We can prove: Theorem 1 If Ḡ is a group of intermediate growth satisfying a law, then

26 We can prove: Theorem 1 If Ḡ is a group of intermediate growth satisfying a law, then Ḡ is not locally graded.

27 We can prove: Theorem 2

28 We can prove: Theorem 2 If Ḡ is a group of intermediate growth satisfying a law, then

29 We can prove: Theorem 2 If Ḡ is a group of intermediate growth satisfying a law, then Ḡ satisfies a positive law.

30 We can also prove Theorem 3

31 We can also prove Theorem 3 There is a group of intermediate growth satisfying a non-trivial law if and only if

32 We can also prove Theorem 3 There is a group of intermediate growth satisfying a non-trivial law if and only if there is a simple group of intermediate growth satisfying this law.

33 THANK YOU FOR ATTENTION