Gemara

71a (Rav Kahana): If a Cheresh (deaf person) can speak (communicate) through writing, we write and give a Get to his wife if he requests this.

Question (Rav Yosef): We already know this from our Mishnah! (If a man became mute and was asked if we should write a Get and he nodded, we test him. If he answers properly no and yes, we write and give the Get.)

Answer (R. Zeira): The Mishnah refers to a mute. He is not like a (total) Cheresh:

(Beraisa): One who can speak but not hear is a (partial) Cheresh. One who can hear but not speak is a mute. Both of them are like Pikchim (sound-minded people) in every respect.

Question (against Rav Kahana - Beraisa): For Metaltelim we rely on gestures and Kefitzah (lip movements) of a Cheresh and what he writes, but not for Gitin.

Answer: Tana'im argue about whether or not he can divorce:

(Seifa - R. Shimon ben Gamliel): This applies only to one who was deaf from birth. If a hearing man became deaf, he writes (a Get) and witnesses sign it.

(R. Yochanan): R. Shimon ben Gamliel argues with Chachamim (the first Tana of his Beraisa. Chachamim say that if a Pike'ach married and became deaf, he cannot divorce even if he can write.)

Support (Abaye - Mishnah): If she went insane, he may not divorce her. If he went deaf or insane, he may never divorce her.

Suggestion: Never means, even if he can communicate through writing!

Rejection (Rav Papa): Really, R. Shimon ben Gamliel explains the first Tana. 'Never' includes even if he is very sharp. Or, 'never' alludes to R. Yitzchak's law:

(R. Yitzchak): Mid'Oraisa, one may divorce an insane woman. Mid'Rabanan one may not, lest men freely have relations with her. (If he went deaf or insane, he may never divorce her, i.e. even mid'Oraisa.)

72a (Beraisa): If a scribe wrote a Get and witnesses signed it Lishmah, it is Batel unless the scribe and witnesses heard the husband's voice command them.

'The husband's voice' teaches unlike Rav Kahana (71a, who said that if a Cheresh can communicate by writing, we write and give a Get to his wife).

Yevamos 112b (Mishnah): If a Cheresh married a Pikachas, or vice-versa, he can divorce her if he wants. He married her by gesturing, and may divorce similarly.

113b - Question (Rav Ashi): Does R. Elazar say that a Cheresh is a Safek Pike'ach because he hold that a Cheresh always has the same weak Da'as (understanding), and he is unsure whether or not his Da'as is clear? If so, his Get is as valid as his Kidushin

Or, perhaps he is sure that his Da'as is weak and (normally) unclear, but at times it is clear? If so, he can be Mekadesh but cannot divorce. (Perhaps his Da'as was clear at the time of Kidushin, but unclear when he divorced!)

Rishonim

Rif and Rosh (Gitin 34a and 6:3): For Metaltelim we rely on gestures and Kefitzah of a Cheresh and what he writes, but not for Gitin. This refers to a Pike'ach who became deaf. If a Cheresh married, we check him. If he wants, we divorce his wife. If a Cheresh married a Pikachas or vice-versa, he can divorce her through gesturing, like he married her. If a Pike'ach married a Pikachas and she went deaf, he may divorce her, but not if she went insane, or he went deaf or insane.

Rambam (Hilchos Ishus 2:26): Whenever we discuss a Cheresh, this is one who cannot hear or speak. If he can do either of these he is like a regular person.

Rambam (Hilchos Gerushin 2:16): If a man became mute but his mind is sound, and he nodded when asked if we should divorce his wife, we test him. If he answers properly, we divorce her. One who became mute is not like a Cheresh.

Rif and Rosh (Gitin 33a and 7:19): A Beraisa teaches unlike Rav Kahana, who said that if a Cheresh can 'speak' by writing, we write and give a Get to his wife.

Kesef Mishneh: The Gemara (71a) connotes that if he can hear or speak, he can authorize a Get through writing. It seems that the Rambam agrees that we rely on gestures and writing of anyone except for a Cheresh, who is like a lunatic. The Rambam does not discuss one who can speak but not hear. It seems that since he is not a total Cheresh, surely he is like a mute.

Gra (EH 120:15): Rav Kahana holds like R. Shimon ben Gamliel. The Beraisa unlike him shows that Chachamim argue with R. Shimon ben Gamliel, like R. Yochanan taught.

Rambam (17): If a Pike'ach married and became a Cheresh or lunatic, he cannot divorce until he gets healthy. We do not rely on his gestures or writing, even if his mind is sound. If a Cheresh married, he can divorce by gesturing. His Kidushin is not mid'Oraisa. Just like he married through gesturing, he may divorce similarly.

Poskim

Shulchan Aruch (EH 120:5): A scribe and witnesses may not write and sign a Get unless they hear from the husband, whether he is Pike'ach, became mute, cannot hear, and whether or not he can speak. Some permit to write and sign if he became mute and commands through writing, and his mind is settled.

Question: The Tur disqualifies even if the husband is a Cheresh. It is a bigger Chidush to disqualify if he is a Pike'ach!

Answer (Beis Yosef DH u'Mihu): One might have thought that we are lenient for a Cheresh, lest his wife be an Agunah (unable to remarry).

Shulchan Aruch (121:6): We rely on gestures to divorce only for one who became mute, or for a mute who can hear, but not for a total Cheresh if he married when he was Pike'ach, or if a Yevamah fell to him from a Pike'ach brother. If a Cheresh married, he can divorce through gestures even if she is a Pikachas.

Question (Beis Yosef DH v'Davka): A mute is one who never spoke, from infancy. We equate him with one who became unable to speak (Gitin 71a).

Beis Yosef (DH u'Mah): The Yerushalmi says that 'just like he married her through gesturing, he may divorce by gesturing' applies to a Cheresh who was Mekadesh with money, but not to Kidushei Bi'ah. The Nimukei Yosef (Yevamos 41a DH Cheresh) brings this, but the Poskim do not.

Question (Mishneh l'Melech, Hilchos Gerushin 2:17): R. Chayim Kimchi questioned the Yerushalmi from a Mishnah (Yevamos 112b). If two brothers were Chershim and one did Yibum, he may divorce her later. The Bavli (71a) says just like she married through gesturing, the Yavam can divorce similarly!

Answer (Mar'eh ha'Panim, Yevamos Reish Perek 14): If one was Mekadesh with money and later had Bi'ah, he does not intend for Bi'ah to be Kidushin. This is why the Yerushalmi says 'if he was Mekadesh through Bi'ah...', and not 'if he made Nisu'in...' Also when he does Yibum (to a Cheresh's wife), he does not intend for a stronger Kidushin (he just 'replaces' his brother).

Aruch l'Ner (Yevamos 112b DH b'Masnisin): The Gemara distinguishes between whether he was a Cheresh from the beginning or not. It does not distinguish between Kidushin through money or Bi'ah!

Shirei Korban (Reish Perek 14): The Yerushalmi thought that we are unsure whether a Cheresh has Da'as. Chachamim enacted that all his actions, e.g. his Kidushei Kesef, are void, but they did not uproot Kidushei Bi'ah. We conclude that he Vadai has no Da'as. The Mishnah equates two Chereshos wives of one man, i.e. even if he was Mekadesh one through Bi'ah.

Beis Shmu'el (9): A Cheresh cannot make a Shali'ach. This does not prove that Shelichus is not needed to write a Get, for this Get is not mid'Oraisa.

Beis Shmu'el (9): If a lunatic was Mekadesh, became sane, and reverted to lunacy, the Beis Yosef is concerned that he had Bi'ah l'Shem Kidushin while he was sane, i.e. if they were secluded in front of witnesses. Some Poskim are concerned for Kidushin if witnesses know that they are living together.

Beis Yosef (DH Kosav bi'Sshuvas): If one is hard of hearing but he hears when people talk loudly to him, he is not a Cheresh. He is like a mute.

Kaf ha'Chayim (OC 343:23): If a Cheresh was considered a minor and he learned to read lips, he is still a minor, because he cannot hear.