In the mathematical subject of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, the Hanna Neumann conjecture is a statement about the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
of the intersection of two
finitely generated subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s of a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
. The conjecture was posed by
Hanna Neumann
Johanna (Hanna) Neumann (née von Caemmerer; 12 February 1914 – 14 November 1971) was a German-born mathematician who worked on group theory.
Biography
Neumann was born on 12 February 1914 in Lankwitz, Steglitz-Zehlendorf (today a distric ...
in 1957.
[Hanna Neumann. ''On the intersection of finitely generated free groups. Addendum.'' ]Publicationes Mathematicae Debrecen
''Publicationes Mathematicae Debrecen'' is a Hungarian mathematical journal, edited and published in Debrecen, at the Mathematical Institute of the University of Debrecen. It was founded by Alfréd Rényi, Tibor Szele
Tibor Szele (Debrecen, 21 J ...
, vol. 5 (1957), p. 128
In 2011, a strengthened version of the conjecture (see
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
) was proved independently by Joel Friedman
[Joel Friedman]
"Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: With an Appendix by Warren Dicks"
Mem. Amer. Math. Soc., 233 (2015), no. 1100.
and by Igor Mineyev.
[Igor Minevev]
"Submultiplicativity and the Hanna Neumann Conjecture."
Ann. of Math., 175 (2012), no. 1, 393-414.
In 2017, a third proof of the Strengthened Hanna Neumann conjecture, based on homological arguments inspired by
pro-p-group
In mathematics, a pro-''p'' group (for some prime number ''p'') is a profinite group G such that for any open normal subgroup N\triangleleft G the quotient group G/N is a ''p''-group. Note that, as profinite groups are compact, the open subgro ...
considerations, was published by Andrei Jaikin-Zapirain.
History
The subject of the conjecture was originally motivated by a
1954 theorem of Howson who proved that the intersection of any two
finitely generated subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s of a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
is always finitely generated, that is, has finite
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
. In this paper Howson proved that if ''H'' and ''K'' are
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s of a free group ''F''(''X'') of finite ranks ''n'' ≥ 1 and ''m'' ≥ 1 then the rank ''s'' of ''H'' ∩ ''K'' satisfies:
:''s'' − 1 ≤ 2''mn'' − ''m'' − ''n''.
In a 1956 paper
Hanna Neumann
Johanna (Hanna) Neumann (née von Caemmerer; 12 February 1914 – 14 November 1971) was a German-born mathematician who worked on group theory.
Biography
Neumann was born on 12 February 1914 in Lankwitz, Steglitz-Zehlendorf (today a distric ...
improved this bound by showing that :
:''s'' − 1 ≤ 2''mn'' − ''2m'' − ''n''.
In a 1957 addendum,
Hanna Neumann further improved this bound to show that under the above assumptions
:''s'' − 1 ≤ 2(''m'' − 1)(''n'' − 1).
She also conjectured that the factor of 2 in the above inequality is not necessary and that one always has
:''s'' − 1 ≤ (''m'' − 1)(''n'' − 1).
This statement became known as the ''Hanna Neumann conjecture''.
Formal statement
Let ''H'', ''K'' ≤ ''F''(''X'') be two nontrivial finitely generated subgroups of a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
''F''(''X'') and let ''L'' = ''H'' ∩ ''K'' be the intersection of ''H'' and ''K''. The conjecture says that in this case
:rank(''L'') − 1 ≤ (rank(''H'') − 1)(rank(''K'') − 1).
Here for a group ''G'' the quantity rank(''G'') is the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
of ''G'', that is, the smallest size of a
generating set
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...
for ''G''.
Every
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
is known to be
free
Free may refer to:
Concept
* Freedom, having the ability to do something, without having to obey anyone/anything
* Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism
* Emancipate, to procur ...
itself and the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
of a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
is equal to the size of any free basis of that free group.
Strengthened Hanna Neumann conjecture
If ''H'', ''K'' ≤ ''G'' are two subgroups of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
''G'' and if ''a'', ''b'' ∈ ''G'' define the same
double coset In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let be a group, and let and be subgroups. Let act on by left multi ...
''HaK = HbK'' then the
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s ''H'' ∩ ''aKa''
−1 and ''H'' ∩ ''bKb''
−1 are
conjugate in ''G'' and thus have the same
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
. It is known that if ''H'', ''K'' ≤ ''F''(''X'') are
finitely generated subgroups of a finitely generated
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
''F''(''X'') then there exist at most finitely many double coset classes ''HaK'' in ''F''(''X'') such that ''H'' ∩ ''aKa''
−1 ≠ . Suppose that at least one such double coset exists and let ''a''
1,...,''a''
''n'' be all the distinct representatives of such double cosets. The ''strengthened Hanna Neumann conjecture'', formulated by her son
Walter Neumann
Walter David Neumann (born 1 January 1946) is a British mathematician who works in topology, geometric group theory, and singularity theory. He is an emeritus professor at Barnard College, Columbia University. Neumann obtained his Ph.D. under the ...
(1990),
[Walter Neumann. ''On intersections of finitely generated subgroups of free groups.'' Groups–Canberra 1989, pp. 161–170. Lecture Notes in Mathematics, vol. 1456, Springer, Berlin, 1990; ] states that in this situation
:
The strengthened Hanna Neumann conjecture was proved in 2011 by Joel Friedman.
Shortly after, another proof was given by Igor Mineyev.
Partial results and other generalizations
*In 1971 Burns improved Hanna Neumann's 1957 bound and proved that under the same assumptions as in Hanna Neumann's paper one has
:''s'' ≤ 2''mn'' − 3''m'' − 2''n'' + 4.
*In a 1990 paper,
Walter Neumann formulated the strengthened Hanna Neumann conjecture (see statement above).
*
Tardos (1992) established the strengthened Hanna Neumann Conjecture for the case where at least one of the subgroups ''H'' and ''K'' of ''F''(''X'') has rank two. As most other approaches to the Hanna Neumann conjecture, Tardos used the technique of
Stallings subgroup graph
Stalling or Stallings can refer to:
*Meanings derived from the term "stall", see Stall (disambiguation)
People
* Stalling (surname)
* Stallings (surname)
Places
* Stallings, North Carolina, a town in North Carolina
* Stallings Field, an airpo ...
s for analyzing subgroups of free groups and their intersections.
*Warren Dicks (1994) established the equivalence of the strengthened Hanna Neumann conjecture and a graph-theoretic statement that he called the ''amalgamated graph conjecture''.
*Arzhantseva (2000) proved that if ''H'' is a finitely generated subgroup of infinite index in ''F''(''X''), then, in a certain statistical meaning, for a generic finitely generated subgroup
in
, we have ''H'' ∩ ''gKg''
−1 = for all ''g'' in ''F''. Thus, the strengthened Hanna Neumann conjecture holds for every ''H'' and a generic ''K''.
*In 2001 Dicks and
Formanek established the strengthened Hanna Neumann conjecture for the case where at least one of the subgroups ''H'' and ''K'' of ''F''(''X'') has rank at most three.
*Khan (2002) and, independently, Meakin and Weil (2002), showed that the conclusion of the strengthened Hanna Neumann conjecture holds if one of the subgroups ''H'', ''K'' of ''F''(''X'') is ''positively generated'', that is, generated by a finite set of words that involve only elements of ''X'' but not of ''X''
−1 as letters.
*Ivanov and Dicks and Ivanov obtained analogs and generalizations of Hanna Neumann's results for the intersection of
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s ''H'' and ''K'' of a
free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and i ...
of several groups.
*Wise (2005) claimed
''The Coherence of One-Relator Groups with Torsion and the Hanna Neumann Conjecture.''
Bulletin of the London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical ...
, vol. 37 (2005), no. 5, pp. 697–705 that the strengthened Hanna Neumann conjecture implies another long-standing group-theoretic conjecture which says that every one-relator group with torsion is ''coherent'' (that is, every finitely generated subgroup in such a group is finitely presented).
See also
*Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...
References
{{reflist
Group theory
Geometric group theory
Conjectures that have been proved